acta physica slovaca vol. 64 No. 2 & 3, 101 – 216 April ...acta physica slovaca vol. 64 No. 2 & 3, 101 – 216 April & June 2014 OPTICAL METHODS IN EXPERIMENTAL MECHANICS Milan Drzˇ´ık,1∗

acta physica slovaca vol. 64 No. 2 & 3, 101 – 216 April & June 2014

OPTICAL METHODS IN EXPERIMENTAL MECHANICS

Milan Drzık,1∗ Juraj Chlpık∗†∗Department of Applied Optics, International Laser Centre, Ilkovicova 3, 841 04 Bratislava

†INPE, FEI Slovak University of Technology in Bratislava

Received 31 March 2015, in final form 20 May 2015, accepted 21 May 2015

Optical metrology methods are an integral part of experimental mechanics. In this commu-nication a systematic study of a variety of these methods is presented concerning the originand subsequent development over the following period of a few decades. Particularly, theadvancement of coherent light imaging in the field of diffraction optics based measuring pro-cedures is described. Primarily, holographic/speckle interferometry used in surface deforma-tion measurements in deformable body mechanics is treated from the viewpoint of opticalscheme optimization. Topics such as image plane holography, pulsed ruby laser holography,electronic speckle pattern interferometry (ESPI) and double-channel speckle interferometry,hybrid experimental-numerical stress state analysis, light diffraction testing of surface rough-ness are discussed as well as their primary applications. Theoretical fundamentals and con-ditions for realization of each method are shown. The main aim of this study is to pointout the basic features and potential exploitation of physical phenomena which are related tointerference and diffraction of coherent light.

DOI: 10.2478/apsrt-2014-0002

PACS: 42.25.Hz, 42.30.Ms, 42.40.-i, 42.40.Kw, 42.40.My, 62.20.-x, 78.20.hb, 85.85.+j

KEYWORDS:

Experimental mechanics, Holographic interferometry, Speckle inter-ferometry, Electronic speckle pattern interferometry, Double-aperturespeckle interferometry, Pulsed holography, Phase visualization, Micro-interferometry, Vibration analysis

Contents

1 Introduction 103

1E-mail address: [email protected]

101

102 Optical methods in experimental mechanics

2 Holographic interference method 1052.1 Holographic interferometry on transparent objects and objects with specularly

reflected surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062.2 Pulsed laser holographic interferometry . . . . . . . . . . . . . . . . . . . . . . 1122.3 Holographic shearing interferometry . . . . . . . . . . . . . . . . . . . . . . . . 1172.4 Holographic interferometry and photoelasticimetry . . . . . . . . . . . . . . . . 121

2.4.1 Polarisation holography . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.4.2 Interpretation of interference pattern . . . . . . . . . . . . . . . . . . . . 128

2.5 Visualisation of ultrasonic bending wave propagation . . . . . . . . . . . . . . . 1292.6 Interferometrical evaluation of the temperature distribution inside transparency . 1342.7 Membrane thickness mapping by Fizeau interferometry . . . . . . . . . . . . . . 140

3 Speckle interferometry 1433.1 Optical system of holographic/speckle interferometer . . . . . . . . . . . . . . . 143

3.1.1 Experimental application of interferometer . . . . . . . . . . . . . . . . 1573.1.2 Two-channel speckle interferometer . . . . . . . . . . . . . . . . . . . . 160

3.2 Reconstruction of records in speckle interferometry using polychromatic light . . 1613.2.1 Application conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.3 Speckle interferometry by sandwich principle . . . . . . . . . . . . . . . . . . . 1643.4 Speckle interferometry utilizing digitization of image . . . . . . . . . . . . . . . 167

3.4.1 Electronic speckle interferometry . . . . . . . . . . . . . . . . . . . . . 1673.4.2 Double-aperture speckle interferometer with electronic record . . . . . . 1733.4.3 Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . 176

3.5 Diffraction of light on the surface microroughness . . . . . . . . . . . . . . . . . 1763.6 Evaluation of elasto-plastic stresses by surface spectrum analysis . . . . . . . . . 184

3.6.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1843.6.2 Separation of the principal strains . . . . . . . . . . . . . . . . . . . . . 188

4 Laser and optical measurement techniques for testing in microelectronics 1914.1 Specular and diffuse-like reflected surface flatness testing . . . . . . . . . . . . . 1914.2 Residual stress measurement of thin films, membranes and MEMS components . 1994.3 Microinterferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2024.4 Vibration measurement of multilayer membrane-like microcomponents . . . . . 207

4.4.1 Visualization of different mode shapes of thin membranes . . . . . . . . 210

5 Conclusions 214

Acknowledgement 214

References 215

Introduction 103

1 Introduction

The role of experimental mechanics involves the measurement of material parameters and physi-cal-mechanical characteristics, the non-destructive flaw detection tests and development or ad-vancement of computational models as well as design of structures. Optical methods representthe most important methods among of experimental mechanics tools. Up to the 60-ies of thelast century the only one tool practically useful for the purposes of experimental stress analysiswas photoelasticity. Photoelasticity is an optical metrology tool which can be used for engineer-ing purposes. It makes use of stress-induced birefringence in transparent polymeric materialsand provides a relationship between the change in state of stress and strain-induced interfer-ence pattern. Photoelasticity can directly visualize only curves of principal stress difference andmaximum in-plane shear stress.

An important impuls for the further development of optical methods was the advent of thelaser light source. As the lasers with coherence light have become relatively accessible, it hasstimulated the development of applications of laser technology including development of inter-ferometrical measurements. The crucial in this respect was a tendency to apply for this purposeholography and afterward also speckle interferometry. Feasibility to use the holography for in-terferometric measurements was firstly demonstrated in 1965. After the first enthusiasm withobservation of interference fringes resulting from double-exposure holographic record of thetwo states of deformation of the object, the problem was discovered, with correct interpretationof interference maxima and minima. Now, the interferograms were generated also on diffuse-likeobjects, the effect previously unknown for classical interferometry.

In this matter in terms of physical interpretations, work by Alexandrov and Bonc-Brujevicof 1967 [1] became fundamental. Authors announced the hypothesis that the arbitrary pointof diffuse surface reflects light into all the directions, that is it reflects the light also in anychosen direction of observation. Thus, we can use the principle of path and consequently phasedifference, as in the classical interferometry, where it was realizable only in the direction ofspecular reflection or passing through the transparent material. Another important finding wasthat, in the vicinity of the observed point, within the speckle conjugated to him, the light isspatially coherent. Therefore, the light from the area of two reconstructed hologram exposuresmay interfere with each other.

The capability to measure the displacement field on diffuse-like surface is one of the mostimportant properties of holographic interferometry. Unfortunately, in the classic schema of Fres-nel holograms record the path/phase difference evaluation is rather complicated. For each pointof the object, solution of the system of linear equations is required for all the three orthogonalcomponents of displacement. For this reason, further developments in this area were focused onthe development of optical schemes of holographic interferometry with simplified direct opticalseparation of the three displacement vector components. By choosing the selected appropriateillumination angles of the measured object and its observation individual displacement compo-nents can be optically separated.

Typical examples of such appropriate arrangement of recording geometry are speckle inter-ferometric methods. The key point in introduction of speckle interferometry was the publica-tion of experimental results carried out by Burch and Tokarski in 1968 [2]. The experimentsshowed analogy between the diffraction effect on chaotically distributed identical structures andthe diffraction on one of these structures. The photographic plate with two speckle fields shifted


one to another diffracts the laser beam in like manner as a screen with two openings away fromeach other to the same shift.

The methods of holographic interferometry as well as speckle interferometric principle foundtheir place in many variations in solving problems of experimental mechanics, namely in exper-imental model testing evaluating stress/strain state. Applications for measurement of nonlinearmaterial behavior but also for material testing in dynamic conditions was an indispensable partof the genesis of these methods over the past decades.

Now, optical experimental techniques have expanded from basic stress analysis of large struc-tures or models to include the electromechanical analysis of micron-sized mechanical elements.Study of mechanical behavior of Micro Electro Mechanical Systems (MEMS) and microstruc-tures by optical methods is undoubtedly promising approach for both inspection and diagnosticpurposes. At the present state of art, there are various forms of experimental-analytical tech-niques which can be applied to mechanical/thermomechanical testing and characterization ofmicroelectronics materials and components. Laser based and other optical metrology tools inthis field owe much of their existence to noninvasive and contactless nature and also to the abil-ity of easy depiction of microworld. Considering the employment of photoelectric sensing byphotodiodes detectors or by CCD the dynamic periodic as well as one-shot surface motions canbe successfully detected. Soon, such capabilities addressed the experiments to be used for the de-tection of forced vibrations of thin membranes and membrane-like components. The laboratoryevaluation has been carried out to determine the mechanical stress state, frequency characteristicsof thin layers damping properties.

Particular attention has to be dedicated to the optical principles which allow their realizationfor in-situ mode of operation. Conditions for immediate tracking of the process as a rule requirethe optical observation through the glass window from a longer distance or the device has tobe installed and operating inside the vacuum chamber. However, in-situ measurements usuallyprovide valuable information on thermal/mechanical characteristics discovering their evaluation.

The application of the well-known principles likewise the development of new procedureshas proved as a tool to solve many issues connected with mechanical and even thermal character-ization of microcomponents, particularly those of the MEMS. In this aspect, the Laser DopplerVibrometry is shown to be a very perspective technique for such a purpose.

Nevertheless, the category of tasks solvable by optical techniques can be very large andirretrievable role falls to pointwise probing but also to the different variations of wide-field CCDbased optical techniques. Thus, the specific features of optical sensing make use of the optics stillpromising in the field of experimetal mechanics not only in solution of variety “non-standard”problems but also MEMS and microelectronics technologies applications.

Holographic interference method 105

2 Holographic interference method

The problem of complete assessment of 3-D displacement vector in holographic interferometrybelonged to the most important and spotted problems of optical interference measurements. Oneof the unique properties of holographic interferometry is the possibility of phase comparison ofwavefronts diffusely scattered off the body surface generally with 3-D shape. In the double-exposure hologram interferometry where the holographic plate is exposed twice, the changesof object induced between two exposures are visualized due to loading by mechanical stress,acoustic loading, temperature deformation and temperature or pressure change of transparentmaterial refractive index.

Using the Fresnel type holographic scheme for double-exposure recording, the obtained dis-placements interference fringes e.g. provide information on surface points displacements withoutsimplifying assumptions. The holographic record of interference fringes by conventional photo-graphic procedure is a known and well-established method. In spite of that the double-exposureholographic interferograms mostly are limited into qualitative description of the studied problemshowing that the quantitative determination of the three orthogonal components of the 3-D dis-placement vector is a troublesome and complicated task. The main reason is that the interferencefringes formed provide information about the motion of object along the sensitivity vector. Ateach point of the object’s surface the separated interference fringes include common informationabout all three orthogonal components. The separation of the components has to be carried outindependently point by point by appropriate selection of three different directions of illuminationand observation, respectively. Interference records evaluation by standard procedure came to beinaccurate in the case when one or two of displacement components are quite different comparedto the others. The work of Alexandrov and Bonc-Brujevic [1] was crucial from the viewpoint ofphysical interpretation of the formed fringe patterns. In the work there was a hypothesis provedabout the diffraction of light reflected off the diffuse-like surface as an optical phenomenon whereevery point on the surface scatters the light in each of spatial directions. It means that we canuse the “standard” interferometrical principle of path difference determination in each of chosendirections. Another important physical factor is observation that in the vicinity of the observedpoint the scattered light is spatially coherent within the appertaining laser speckle, consequently,the light reconstructed from both the hologram exposures can interfere with one another. Thementioned notes lead to the possibility of quantitative evaluation of paths differences for all thepoints covered by interference pattern.

Nevertheless, to do it, we have to solve a system of three linear algebraic equations for eachof the points. Such an evaluation is time consuming, hence in seventhies, an effort to propose theoptical scheme where the displacement components in a certain coordinate system considerablysimplify the whole procedure of object deformation description emerged as a basic tendencyof experimental studies. As a rule this demand called for direct optical separation of in-planeand out-of plane displacements. In experimental practice the description of the double exposureinterferograms was based on the choice of suitable conditions of object illumination as well asangles of observation. Even today, there is not yet a simple and reliable quantitative methodto obtain information on spatial displacements on objects. Fig. 2.1 illustrates the Fresnel typedouble-exposure holographic interferogram observed from two different angles of observationand illumination.

One of the unique properties of the lens is the feasibility to separate the spatial frequencies.


Fig. 2.1. Double-exposure Fresnel type holographic interferogram of the gun deformation between twosubsequent shots. The interference fringes represent isolines of equal magnitude of deformations.

Using a lens to image a given object near the hologram recording plane creates a hologram inwhich the image of the object can be located near the hologram plane. Such an image-planehologram has some specific features, one of them is the peculiarity of the reconstruction bywhite light illumination. However, when the developed hologram is illuminated at the recon-struction process we cannot be able to observe the image of the object immediately. Instead, theimage restores only the lens aperture viewing angle through which the image-plane hologramwas recorded. The reason is that at reconstruction we also reconstruct the quadratic phase fac-tor which is associated with the imaging conditions. A real image of the object can be wievedalong the optical axis of the image hologram recording scheme by reversing the reference beamthrough the hologram which needs an additional objective lens. It can be approved, when recon-structing the image-plane hologram, both the real as well as virtual images of the object occur atthe same place, where the original real image of the object created by recording lens was situ-ated. For interferometry use the most important property of such a hologram is the confinementof observation angles at the double exposure hologram recording by recording lens aperture.

Despite the relatively small effort in the field, the use of image-plane holography for holo-graphic interferometry purposes was a promising trend namely in experimental mechanics. Thetendency to separate the in-plane displacement components can be documented also by develop-ment of so-called holographic moire-interferometry, and namely variety of schemes of speckleinterferometry but also combined methods based on image-plane holography which were carriedout the following years.

2.1 Holographic interferometry on transparent objects and objects with specularlyreflected surfaces

In classical interferometry two smooth wavefronts interact with each other because of differencesin their path lengths. With regard that the appropriate temporal as well as spatial coherencies areneeded to obtain a static interference pattern, the interfering wavefronts have to be reflectedfrom glossy surfaces. As a result, classical interferometry is limited to measurements of onlyspecularly reflected or nondiffusing transparent flat surfaces where the angles of illumination


Fig. 2.2. Optical scheme of holographic interferometer for transparent object deformation measurement.Md – semitransparent mirror, bemasplitter, M – miror, P1 – collimator pinhole, C1 – collimating lens,M0 – transparent object, L1 – well corrected imaging lens, P2 – reference beam pinhole spatial filter, H –holographic plate.

and observation are limited by mirror-like reflection. Thus, there is no possibility to obtaininformation e.g. on in-plane displacements.

In the field of experimental mechanics, where the study of mechanical stress state is of pri-mary importantance, model experiments are frequently utilised. For a long time a typical exam-ple was an application of interferometry to study the deformation of loaded planar models in thestate of plane stress or strain conditions. Using the transparent models (made of e.g. plexiglassor polymeric resins), the transversal deformation has been interferometrically measured. Fromthe theory of elasticity, the resulting interference fringes can be simply interpreted. In manyapplications it is not necessary to watch the information on in-plane surface displacement com-ponents. When a non-diffusing object with mirror-like surface or transparency is illuminatedfrom behind by a koherent wavefront and the hologram is then made by interferency with ref-erence beam, the reconstructed image will contain no speckle structure. Therefore, an opticalscheme of image-plane holography can be adjusted, where the holographic interferometer ac-quires several specific features. In order to study in detail the state of stress in planar modelsmade of polymeric transparent materials, we have developed the optical setup as it is drawn inscheme of Fig. 2.2. The coherent beam of laser light is divided by a semitransparent mirror Md

into two parts. For optimal adjusting, the continually variable mirror beamsplitter has been used.The laser sources used were mostly He-Ne CW lasers with output power 40 mW to 60 mW withcoherent length of 20 cm to 80 cm. In the schematics P1 and P2 are the spatial filters of pinholes.Spatial filters are the point-like sources of both the information and the reference waves. Theplane wave of a larger diameter created by a collimator C1 is passing through the studied model.L1 denotes the imaging set which is projecting the model image into the plane of holographicplate H . To record the holographic scenes with transversal dimensions of several centimeters,the well corrected objective lenses were used with focal distance 100 mm to 200 mm and theaperture number of 1/2.8. The setup shown does not use any ground glass in the information


Fig. 2.3. Double-passed holographic interferometer for transparent object deformation measurement pro-viding twofold sensitivity. P – point-like light source pinhole, C – collimating objective lens, Md – semi-transparent mirror, M , – information wave mierror, MR – reference wave mirror, M0 – transparent object,L1 – well corrected imaging objective lens, H – holographic plate.

wave, thus, no speckle effect is visible in the hologram. Another alternative of the groundglassfree scheme is drawn in Fig. 2.3. In this case, the information beam is created after the reflectionfrom semitransparent mirrorMd and its subsequent passing through the transparent model. Afterpassing through, the beam is reflected again from the flat mirror MR, thus, the double passingof the wave through model is realized. Compared with the previous scheme, the setup providesfavourable twofold sensitivity of the model thickness measurement.

The optical ray tracing in both the schemes requires the diameter of the imaging lens tobe slightly larger then the measured area of the model. Moreover, care must be taken whenthe observed planar model gradients of thickness variations are two strong and the rays passingthrough the transparency are deviated beyond the imaging lens boundaries. In mechanics, thepoints with stress singularities (e.g. crack tip or load points) are often the case. Lack of specklestructure in this type of holograms offers for such a strong stress gradients another benefit in thepossible big enlargement of the recorded object images-interferograms. These interferogramsthus, can be postprocessed and studied in detail point by point. The maximum magnification islimited only by the resolving power of the optical system of the holographic recording.

Holograms generated in the image plane of the imaging system rank in a specific class of theholographic records with a number of positive properties when using it as an interferometricaltool. The lack of speckle structure can be revealed in microscope where only a system of densefine fringes is visible. Note that to view the grating, a microscopic objective with a relativelystrong aperture number has to be used, to catch the angle larger than the angle between theinformation beam and the reference beam.

During hologram reconstruction we have interference fringes localized in the plane of holo-gram which are in fact the same as moire fringes originated by overlapping both the fine mi-croscopic interference gratings belonging to two holograms of the double exposure. When thehologram is reconstructed, the positions of gratings intersections do not diffract the light effec-


−1.

+1.

0Source

Lens

CCD

Fourier transform plane

Diaphragm

Fig. 2.4. Schematic of image-plane hologram reconstruction using point-like light source.

tively, hence they correspond to interference minima of the fringe pattern. The coincidence ofthe plane of interference fringes localization with the hologram plane allows us to use for recon-struction the source with weak coherency, actually even the white light extended sources basedon incadescent lamps can be used. This is one of the unique features of all the image-planeholograms.

However, the reconstruction procedure especially when not large but the point-like lightsource is used, has to be modified in order to compensate the quadratic phase factor existingin the plane of hologram. Illuminating of the hologram by a replica of the reference beam willreconstruct only the lens through which we may observe the object. The reason is that we recon-struct also the quadratic phase factor associated with the imaging condition. The use of imaginglens to project the object image into optically conjugated image plane transforms the parallelwave into convergent spherical wavefront and the image formation is described by a complexamplitude where the phase factor of the spherical wavefront is present. At reconstruction, whenthe developed hologram is illuminated by a complex conjugate wavefront of the reference wave,we can observe a real image of the object full-area of which is illuminated. In practice, the mostconvenient way is to set a scheme as in Fig. 2.4.

Double exposure holograms obtained by optical setup in Fig. 2.2 or Fig. 2.3 give the interfer-ence pattern with infinite interference fringes which are analogous to fringes produced by classicinterferometry principles. In principle, also finite fringe pattern can be obtained if one of theholographic interferometer mirrors is slightly tilted between exposures.

Nevertheless, the fundamental benefit of the holographic record is its differential nature,where the two true shapes of the object wavefronts are each other compared. Seeing that thefringes at double exposure are created as an optical paths difference between the real not anideally planparalel transparent or a flat surface unloaded model shape with the shape loadedand deformed, the fundamental problem of classic interferometry in experimental mechanics iscompletely avoided.

Interference fringes on the planar models are created as an optical path difference betweenthe state of model thickness before and after the loading deformation. Intensity distribution oftwo interfering waves is described by the well-known expression

I(x, y) = 2a2

[1 + cos

2πλ

(l2(x, y)− l1(x, y)

)], (2.1)


where a is the amplitude of both the interfering waves l1(x, y), l2(x, y) and λ is the light wave-length. The path difference in the transparent model with initial thickness h and the refractiveindex n0 can be expressed

l2(x, y)− l1(x, y) = h∆n− (n0 − 1)∆h , (2.2)

where we have omitted the term dn0 dh as a negligible value, and the refractive index of theambient air is equal 1. The basic relationship to interpret the interference pattern follows fromthe elasticity equation for transverse deformation at the state of plane stress

∆h = −νhE

(σ1 + σ2) , (2.3)

where σ1 +σ2 is the sum of principal stresses and E and ν are the material parameters – Young’smodulus and Poisson’s ratio, respectively. If we assume optically isotropic transparent mate-rial with negligible birefrigence properties (e.g. plexiglass), the proportionality follows fromMaxwell-Neumann law between the sum of principal stresses and the change of refractive indexinduced by loading

∆n = C0(σ1 + σ2) , (2.4)

where C0 is the optical constant of optically isotropic material. After substituting into Eq. (2.1)and taking into account the interference extreme condition

l2(x, y)− l1(x, y) = Nλ , (2.5)

we obtain the expression for the sum of principal stresses in the form

σ1 + σ2 =λ

C0 + (ν/E)(n0 − 1)· Nh, (2.6)

where N is the interference order. As seen from this expression for interference fringes inter-pretation, the quantitative values of principal stresses depend on fundamental material as wellas optical constants. That is, for every interference fringe order which is in the case of planarmodel under plane stress condition of Eq. (2.3) so-called isopachic fringe, the stress value can beassigned by using the fringe value ch and the thickness of the planar specimen h

σ1 + σ2 = chN

h. (2.7)

The interference sensitivity constant – fringe value is composed of two main parts, materialand optical. For that reason the best way how to determine this coefficient for assigning stressesto the measured deformation is the calibration experiment on the specimen with known stressdistribution. The static value of the fringe value for material of polymetylmetacrylat (PMMA)Acrylon has been measured by well defined loading of plexiglass beam with dimensions 300 ×45×6 mm3 in pure bending. From density of interference fringes of isopachics on the plexiglass(PMMA) beam the fringe value ch = 8.7 × 103 Nm−1 at the wavelength λ = 633 nm wasdetermined.


h

ɑ

L

P

P/2 P/2

Fig. 2.5. The specimen of PMMA beam with simulated crack.

(a) (b)

Fig. 2.6. Interference fringes of equal thickness change around the loaded crack, (a) crack perpedicular tothe beam edge loaded by bending, (b) crack inclined to the load direction in the beam constantly loaded intension.

The essential attribute of the holographic interferometry success is the comparative workingprinciple which takes off the necessity of using precise optics and planarity or flatness of themeasured models. In the last thirty or forty years in the field of mechanics, an intensive attentionhas been devoted to study the cirkumstances of the mechanical failures caused by the presence ofcracks. We have proposed and carried out a number of applications to study such singular stressconcentrators where cracked specimens made of plexiglass transparent material have been me-chanically loaded and the stress distribution was evaluated. Experimentally the interferometricpatterns of isopachics (see Eq. (2.7)) have been recorded in the setup schematically outlined inFig. 2.2. For better imagination, one of the models with simulated crack is depicted in Fig. 2.5,also models of plates in tension have been measured. The specimen thickness varied for differ-ent specimens between 5 mm to 10 mm, for measurement in immersion liquid the thickness of16 mm was used. The thickness of the loaded beams was chosen from the requirement of fulfilledplane stress conditions at once with enough interferometrical sensitivity of fringes formation. Anexample of the typical interference patterns around the crack tip under bending loading of thebeam are in Fig. 2.6 where the loading of perpendicular and also inclined crack is visible.

As it was explained above, see Eq. (2.1) to Eq. (2.6), the comparison of two wavefronts


passing through the unloaded model and through the model after transversal deformation, a lightfield distribution with the intensity I is obtained by simple alteration of these expressions

I ∼ cos2[πλ

(C0 +

ν

E(n0 − 1)

)hS], (2.8)

where by S the sum of principal stresses was denoted. Knowing the functional dependence ofindividual components in analytical expression or from numerical modeling of the mechanicalproblem, the invariant of principal stresses can be simply experimentally evaluated. Particularly,in fracture mechanics the analytical solution of the stress state in a vicinity of crack was expressedby [3] in form of infinite series. For tensile mode of crack loading it holds

σx = σxs +O(r) ,σy = σys +O(r) ,

(2.9)

where σxs, σys are the singular terms reciprocally related to square root of radial distance fromthe crack tip [4] and O(r) denotes polynomial series of the order r. Assuming that the fringepattern formation is governed by the asymptotic stress field near the crack tip, the geometry ofthe fringe pattern can be expressed according to Eq. (2.7) by the summation of principal stresses

S = chN

h=

2√2πr

KI cosθ

2+O(r) , (2.10)

where θ is the polar angle of the point with respect to crack paths line, r is the polar distancefrom the crack tip and KI is the stress intensity factor of the first mode of crack loading. Whenthe crack walls are loaded also in shearing mode

S =2√2πr

(KI cos

θ

2−KII sin

θ

2

)+O(r) , (2.11)

where KII is the stress intensity factor of the second mode of crack loading. The stress intensityfactors are a function of the applied load, the geometry of the specimen, and the length of thecrack. For quantitative evaluation they have to be determined by numerical simulation.

As it is clearly seen in the examples of the fringe pattern, in the area of crack tip, the fringesare too dense and in these figures even unresolvable. Nevertheless, if a wide open objective lenswith better numerical aperture is used to photograph the reconstructed image, we can enlarge thecrack tip vicinity in detail. Experimental technique used allows to observe the thickness changesup to the distance of some tenths of mm from the crack tip. In Fig. 2.7 even shape of small plasticzone is visible.

2.2 Pulsed laser holographic interferometry

Investigation of dynamic processes in mechanics is one of the most complicated tasks of exper-imental mechanics. Holographic interferometry, using pulse laser as a coherent source, can beregarded as an effective tool to study a variety of dynamic phenomena. The primary requirementfor a pulse holography is the laser source with increased both spatial and temporal coherenciesand with the pulse time dependence in the range of at least nanoseconds. In the experimentswe used the laser with ruby crystal and passive Q-switch. The energy of one flash was about0.1 J without modulation or 0.025 J with application of Q-switch. In this series of experiments,


Fig. 2.7. Enlarged detail of the crack tip with plastic zone. Apple-like shape of the fringes is clearlydeformed.

the problem of stress distribution at fracture of the fast propagating crack was studied. Theo-retical solutions cannot completely explain some of the effects, such as non-elastic deformation,accumulation of micro-defects, discontinuity of the cracking process etc. The local flaws in thefracture process zone as well as the properties of material play a significant role, thus, the resultsof the interferometry with high spatial resolving power measuring close vicinity of the runningcrack tip could be a useful addition. As previously on planar transparent models, to observeisopachics interference fringes, the experiments were carried out by the optical arrangement ofan interferometer, where the diffusor screen is not included, and the recorded interference pat-tern can be considerably magnified. The double-exposure holograms were taken by a pulsedruby laser with the output power of ∼ 50 mJ at wavelength λ = 694.3 nm. The lack of groundscreen enables to arrange correlative matching parts of the viewfield area of both the referenceand information beams in order to be overlapping on the holographic plate. Any possible imper-fections in spatial coherency of the laser beam can thus be eliminated. To adjust such an opticalsetup, one flat mirror and converging lens were introduced in the reference beam which upturnedthe beam. As a result, the diffraction sensitivity of holographic records was excellent even with-out using any bleaching process. In addition to application of color filter KC-19 as a passiveQ-switch, the ruby laser was put to use in automodulation mode. As known [5,6] at the thresholdgeneration even Q-switch-free laser resonator generates only one or a few successive pulses. Thetypical width of these peaks was about 100 ns with 8 µs to 15 µs time intervals between possibleensuing pulses. These laser pulses show a good coherency. On the other hand, automodulationmode arises as a result of thermal deformations of the ruby crystal, thus, the main drawback ofthis mode of operation is its considerable unstability.

The catastrophic propagation of cracks is observed on the three-point bending specimens.These slender beams with the thickness of 4 mm were cut from the sheets of plexiglass (PMMA).Crack movement starts from an initially short blunt notch by the acting loading force generatedafter releasing a preliminary compressed strong coil spring. Mechanical releasing of the springinitiates not only the external load, but also triggers the countdown in the synchronisation blockof the laser. Time delay circuit also reads a scheduled time interval in the range of 20 µs to


0.2 0.4 1.0 1.40

0.6 0.8 1.2 1.6

100

200

300

0

0.2

0.4

0.6

0.8

1.0

1.2

Timetms

Load

ingt

For

cetN

tK

ID ttMP

atm1/2

cracktpropagation

1.4

Fig. 2.8. The determined values of dynamic stress intensity factors KID (dashed line) determined fromisopachics data and the time development of the loading force P (solid line).

Fig. 2.9. Instantaneous dynamic deformation in a beam made of PMMA at the moment of cracking.

1200 µs during the crack propagation. Before that, the first exposure of the hologram is carriedout at unloaded specimen. The sequence diagram is as follows: After triggering, the externalload increases as shown in Fig. 2.8, where the time history of the load for the whole beam crack-ing process is drawn. It can be seen that the loading force grows monotonically and passes acritical region when the crack starts to propagate. The loading time to initiation of the fracturetakes about 1 ms, that is the quasi-static conditions before rising of the crack are guaranteed andthe influence of the stress waves reflected from boundaries of the specimen is negligible. Af-ter a time interval, the crack reaches a position at which the second laser pulse is flashed. Atthis second exposure, interference fringes of an instantaneous deformation around the crack arerecorded. Fig. 2.9 shows the fringes in the region of propagating crack where also the instanta-neuos deformation around point like load is visible. Crack tip speed at the moment of the secondexposure is determined by reading on an oscilloscope screen of time intervals between breakingof conductive graphite lines drawn on the beam surface.


In Fig. 2.8 the development of the stress intensity factor value determined individually forseveral cracks with different propagation velocities is also plotted. The values of stress inten-sity factors are determined by analysing of isopachics fringes according to the above describedmethod based on sum of principal stress determination for plane stress condition. The catas-trophic crack growth is a fast dynamic process where the velocity of stress waves in solid mate-rial could be important. To evaluate the dynamic stress state, two main factors have to be pointedout. If the crack tip propagates very fast, the velocity of stress wave cannot be regarded as infiniteand we have to study the equations for elastic body with terms which define the dynamic inertialinfluence. The value of the first invariant, under the assumption of constant crack tip velocity, isfound by [7] and [8] from the relationship

σx + σy =(s22 − s21

)[φ1(z1) + φ∗1(z1)] , (2.12)

where φ1(z1) is the complex potential function, and

sj =[1− (a/cj)

2]−1/2

, j = 1, 2 , (2.13)

where c2, c1 are the velocities of shearing and longitudinal stress waves, a is the crack tip ve-locity. Assuming the complex stress function φ1(z1) in the form of series we get expression forstress intensity factor in dynamic propagation conditions as it was defined by [7]

KID =√

2π4s1s2 −

(1 + s22

)21 + s22

A1 . (2.14)

After a short analysis of this expression, when real values of maximum crack tip velocities areregarded i.e. the value of a/cR does not exceed 0.2 to 0.3, then a very small dynamic correctionfactor of a few percent can be found. As a rule, such a small dynamic correction is of the order ofexperimental uncertainty and of measurement errors. The strong viscoelastic properties of poly-mers including such as PMMA, are much more important factor. Also PMMA presents strongstrain rate dependence. An assessment of the influence of viscoelasticity on the observed strainfield is experimentally complicated particularly due to change of the fringe value ch describedby Eq. (2.6) and (2.7). For dynamic conditions this fringe value must be corrected.

The concept of linear viscoelasticity is based on the approximate relationship between thetime dependences of the stress and the strain values. For a stress gradient of amplitude σ0 whichoscilates sinusoidally with a constant frequency ω,

σ(t) = σ0 exp(iωt) , (2.15)

the material response can be written in the form

ε(t) = ε0 exp[i(ωt− ϕ)] , (2.16)

where ϕ is the phase angle of time delay of ε(t) after σ(t). The effective complex modulus ofthe material is defined as

E = E1 + iE2 =σ0

ε0exp(iϕ) . (2.17)


10 mm

Fig. 2.10. Instantaneous dynamic deformation in the near vicinity of crack tip running with velocity400m s−1 at the moment of cracking.

As a result, the constant phase shift over the whole field causes that the strain field is followedby the same spatial stress distribution, consequently, the fast propagating crack tip is situated ina medium which seems to be increasingly stiff. This effect is clearly identified by a simpleanalysis of the shape and spatial distribution of transversal deformation around the near vicinityof a running crack tip as seen in one of the recorded examples of the beam cracking (Fig. 2.10).This is the case of crack tip velocity of about 400 ms−1. As it is demonstrated when we aremoving along the crack path, the layout of individual fringe curves shows the shift of the curvesone after the other in the crack propagation direction. The curves, and therefore the deformation,further away from the point of crack tip get behind those of close to the crack tip. Moreover, asthere is an opportunity to enlarge the hologram to the considerable extent, a scene at the nearestcrack tip vicinity can be closely viewed. The immediate region is noticeable, although fringesare slightly blurred due to finite exposure time 60 ns to 80 ns of the laser flash pulse.

The quantitative stress-strain relation is described by the complex modulus of material forthe given frequency of the oscillation or loading pulse. Then the fringe value ch of Eq. (2.7) canbe corrected knowing the quantitative changes of this modulus. The deviations of the parametersλ, n0 and C0 as well as Poisson’s ratio in the situation of fast varying loading are negligibleas it is known from many experimental measurements. Hence, the most important parameterbecomes the modulus E. Since the dynamic value of Young’s modulus depends on the loadingfrequency, it is necessary to find relationship between the crack tip velocity and the time historyof deformation near the crack path. An assessment has been performed on the assumption thatfor a material near the moving crack path the time dependence of deformation appears as astress pulse [9]. Provided that the crack speed is nearly constant, the deformation around thefast running crack tip may be used to determine the time history of the material loading pulse.The typical frequency of this pulse is obtained by a spectral analysis. Fig. 2.11 presents theshape of stress pulse for the example of fracture at the velocity of crack tip propagation of about400 ms−1. The spectral content of the pulse is concentrated around the maximum of about41 kHz. The quantitative value of the modulus of elasticity in the region of such frequenciesexceeds 6.18 MPa [10], in comparison with its static value of 3.12 GPa. The dynamic fringevalue for isopachics evaluation exceeds its static counterpart by 43 %. Using the corrected fringevalues true values of dynamic stress intensity factors were determined for different crack tipvelocities. The results are drawn in Fig. 2.8.


5 10 15 20 25 300

5

10

15

20

25

30

Time µs

σx +

σy M

Pa

r/a = 0.05

0.1

0.15ɑ = 400 m/s˙

40

Fig. 2.11. Stress pulse near the crack path produced by a passing by crack tip.

2.3 Holographic shearing interferometry

Shearing interferometer has been utilized for years in the optical industry as one of the testingtools. Its usefulness was approved particularly in testing of aberations of spherical as well as as-pherical optics elements. Besides this, the principle of interferency of mutually shifted identicalwavefronts belongs to optical methods which are being used for measuring deformation of thesurfaces with mirror like reflection of light. This approach has shown to be useful also for theinvestigations in experimental mechanics. Namely the possibility to measure directly the slopecontours formed due to mechanical or thermal load of the initially flat surface is important be-cause of the direct relationship of the surface slopes and the moments, according to the theory ofthin plates (see Fig 2.12). But, the interference patterns obtained by shearing interferometry havebeen exploited in investigation of cracks or various mechanical defects, as well. From the pointof view of the wavefront shear realization there are many ways how the shearing interferometer

Fig. 2.12. Slope contours of the centrally loaded plate obtained by double-exposure holographic shearinginterferometry.


Planewave

Hp q

Δx

M

Referencewave

Informationwave

Fig. 2.13. Optical scheme of holographic shearing interferometry. The shift of object position between twoexposures is ∆x [11].

1 2

Δxn0

n1 n2

S1 S2

Δh1

Δh'

h'h

Δh2

ϱ

Fig. 2.14. Scheme of light passing through a transparent object. Interferency in shearing interferometryis composed of three effects: change of object thickness ∆h with position, change of refractive index ∆ninduced by stress intensity S, deformation of the object planarity.

can be designed. In our laboratory, unlike using of various beamsplitting elements, a holographicvariant of the shearing interferometry has been proposed. Two independent exposures are spacedin time and the wavefront shift is entered by mechanical displacement of model or alternativelyby optical shifting of wavefront, see Fig 2.13. The recorded double exposure hologram providesthe overlapping wavefronts interferency at the reconstruction. The main drawback of the classicinterferometry namely the necessity of the very precise elements in optical setup with wide openviewing fields and the ideal planparallelity of the models can thus be avoided.

The basic idea of the differential (shearing) interferometry is comparison of phase betweentwo neighbouring points of transparent planar model or the points on specularly reflected surface.The shift of the model between both of the exposures is realized by mechanical removal ∆x fromposition 1 to position 2. In Fig. 2.14 the light rays passing through the transparent object areschematically drawn. The object is assumed to be uniformly illuminated by a plane wave. Let anamplitude behind the transparency at a reference plane % is as known

a1(x, y) = a1 exp [ikl(x, y)] (2.18)


in position 1, and

a2(x, y) = a2 exp[ikl(x+ ∆x, y)] (2.19)

in position 2. In these expressions l(x, y) and l(x+ ∆x, y) are the optical paths in points 1 and2, respectively, and ∆x is the displacement between both of the rays. In the case of aligning boththe rays by shiffting of wavefronts the interferency is created and thus the intensity distributionis following

I(x, y) = 2a2[1 + cos k

(l(x+ ∆x, y)− l(x, y)

)], (2.20)

where a1 = a2 = a. For interference maxima applies

2πλ

[f(x+ ∆x, y)− f(x, y)] = 2πN , (2.21)

where N = 0,±1,±2 . . . is the fringe order. The Eq. (2.21) is a common expression for inter-ference fringes interpretation in shearing interferometry.

Next, let us next consider the optical path difference for planar transparent model in a planestress state. For phase in positions 1 and 2 is

l(x, y) = (n+ ∆n1)(h+ ∆h1) + n0∆h′ (2.22)

or

l(x+ ∆x, y) = (n+ ∆n2)(h+ ∆h2) , (2.23)

respectively. In the expressions h is the initial model thickness, ∆h1 is the change of this thick-ness originated by load in position 1 and likewise ∆h2 in position 2, n is the refractive indexof the unloaded transparent material, ∆n1,∆n2 are the changes due to loading. Assuming aspreviously, the validity of Maxwell-Neumann law for optically isotropic material, the refractiveindices changes ∆n1 and ∆n2 induced by load are related as Eq. 2.4. Then for differences inboth the points 1 and 2 there is the same linear relationship

∆n2 −∆n1 = C0(S2 − S1) = C0∆S , (2.24)

where S1, S2 are the sums of principal stresses at corresponding points and ∆S is their dif-ference. In plane stress the change of planar model thickness is described by Eq. 1.51, thenalso

∆h′ = ∆h2 −∆h1 = −νhE

∆S . (2.25)

By subtraction the Eq. (2.22) and Eq. (2.23) we have

l(x+ ∆x, y)− l(x, y) = h(∆n2 −∆n2) + (n− 1)∆h′ , (2.26)

where the small values of ∆n1∆h1, ∆n2∆h2 have been dropped and the refractive index n0 hasbeen assumed to be 1 as it is refractive index of ambient air. Substituting Eq. (2.24) and Eq. (2.25)into Eq. (2.26) and taking into account condition of interference maxima Eq. (2.21), the basicexpression for fringes interpretation is obtained after simple rearrangement of the difference ofprincipal stresses sums in two neighbouring points


Fig. 2.15. Butterfly-like fringes present first derivatives of sum of principal stresses around loaded crack tipin PMMA beam.

∆S = chN

h; ch =

λ

C0 − (ν/E)(n− 1), (2.27)

where ch is the fringe value, the same as in Eq. (2.7) for isopachics fringes. It can be determinedexperimentally similarly applying calibration on the bending beam.

When the value of the wavefront shear is chosen sufficiently small then the approximation ofinfinitesimally small difference can be applied

∂S

∂x=

∆S∆x

(2.28)

or after the substitution into Eq. (2.27) we get

∂S

∂x=

ch∆x

N

h. (2.29)

By analogy for the shifft of the wavefronts in the direction of coordinate axis y

∂S

∂y=

ch∆y

N

h. (2.30)

These simple formulae show the immediate coupling to the fringes of isopachics, in the planarmechanical elements, the fringes can be interpreted as contours of derivatives (slopes) of thesums of principal stresses. In Fig 2.15 there is an example of observed contours of derivativesaround the loaded crack tip in PMMA beam. Their interesting property is the opportunity toadequately control the measurement sensitivity by a proper adjusting of the relative shifft of boththe wavefronts. In mechanics it is often encountered because of the wide range of magnitudes ofdeformation that may occur in experiments. As we have seen from Eq. (2.29) and Eq. (2.30) forfringes emerging the next rule is evident – the larger the shifft the higher the sensitivity. However,it is necessary to note that the range of degree of control is fairly limited. It becomes relevantparticularly when we are dealing with not very precipitous thickness variations or slopes, whenthe density of fringe pattern is small and the larger difference of wavefronts positions has tobe predetermined. On the other hand, it leads to greater uncertainty, and then we have poorlydefined positions of points in object coordinate system.


Another limitation of the shearing interferometer is its working principle where the doublewavefront is compared by its own. The records of both the first and the second exposures bearinformation only on true thickness of object which as a rule in mechanical experiments is notperfectly planparallel and some kind of deformation exists. In mechanics problems for the mostpart, the slopes contours of deformation formed by load are dominant and some distortions due tononparallel model are visible at once. Particularly, when dealing with singularity problems, thiseffect can be intentioned as a second-rare. Generally speaking, the method involves very simpleholographic setup, when compared to other interferometric techniques, it is relatively insensitiveto vibrations. Insignificant sensitivity of the shearing principle to rigid body motions, especiallyin experiments where the mechanical loads are applied, cannot be ignored, too.

The case when the information wavefront is reflected off the mirror-like surface the pathdifference is described simply as

l(x+ ∆x, y)− l(x, y) = 2(∆h2 −∆h1) , (2.31)

where the factor of two in front of the bracket means double passing of the light rays to thesurface and back. Plane stress condition of loading leads to the next expressions for emergentinterference fringes

∂S

∂x= − λE

ν∆xN

h,

∂S

∂y= − λE

ν∆yN

h.

(2.32)

2.4 Holographic interferometry and photoelasticimetry

The material contribution to the fringe value can be determined also by an arrangement of theexperiment not for passing the light through the transparent model but with reflection on one sidewall of the beam. This is the case, where no contribution of optical fringe value C0 is observedand the fringe value ch can then be obtained by simple calculation from the known materialparameters E and ν of the PMMA Acrylon

ch =λ

(ν/E)(n0 − 1), (2.33)

where for Acrylon E = 3.12 GPa, ν = 0.286 [10] refractive index at the conditions of roomtemperature is n0 = 1.497. Unfortunately, the fringe value calculated by such a way is notalways in coincidence with the “material” fringe value obtained experimentally which is morereliable. This fact is often ignored and the influence of C0 on the interference optical path isconsidered as unimportant.

The measurement as we have proposed for optical constant c0 determination seems to bemuch more realistic. The transparent model is immersed in a liquid in a special cuvette (Fig. 2.16).The defined load deforms the beam and the inside deformation of the transparent material inducesa small but in interferometry significant change of refractive index according to linear relation-ship of Eq. (2.4). Considering negligible difference between the refractive index of immersionliquid (e.g. glycerol) and the transparent model material, in the optical scheme with both sides of


Planewave

Load P

Model

Immerseliquid

n0

n0 n1,2

Fig. 2.16. Passing of the information wave through the transparency in immersion liquid.

beam in immersion liquid, the transversal change of the model thickness has no effect on the totalpath difference. For that reason, the interference fringes are created only due to stress inducedchanges of refractive index. As an example, the calibrating experiment of the Acrylon materialgave the fringe value ch = λ/C0 = 22.4× 103 N m−1 at wavelength of λ = 633 nm, which canbe assumed as a constant for transparent material.

The concept of measurement of the transparent planar model placed in the immersion liquidcan be successfully put to use also in the case when we are dealing with comparatively thickmodel, when the state of stress is better described by elasticity equations as a plane strain. Gen-erally speaking, plain strain conditions better represent also the state of stress near the free edgesat the contour of the planar models.

In the 19th century Maxwell discovered that for a linearly elastic material, the changes inthe indices of refraction are linearly proportional to the principal stresses and refractive indexellipsoid is considered coaxial to the stress or strain ellipsoid. For the general three dimensionalcase, the refractive index changes can be written as

n1 = n0 + C1σ1 + C2(σ2 + σ3) ,n2 = n0 + C1σ2 + C2(σ3 + σ1) ,n3 = n0 + C1σ3 + C2(σ1 + σ2) ,

(2.34)

where the C terms represent the stress optic coefficients and the σ terms represent the principalstresses, along the respective axes. The term n0 is the index of refraction of the unstressed mate-rial. As seen, there is variation of refractive index as linear functions of the stress along directionsof principal stresses and the sums of stresses in the perpendicular directions. In particular, forplane stress conditions σ3 = 0 and the system of Eq. (2.34) is simplified

∆n1 = C1

(σ1 +

C2

C1σ2

),

∆n2 = C1

(σ2 +

C2

C1σ1

),

(2.35)

where ∆n1 and ∆n2 are the changes of refractive index in the directions of the refractive el-lipsoid axes. Taking into account optically isotropic material the ellipsoid is transformed intosphere and the stress optic coefficients C1 = C2 = C0, which leads to the Eq. (2.4),

∆n =12(C1 + C2)(σ1 + σ2) . (2.36)


Fig. 2.17. Isopachics on PMMA beam in immersion liquid.

In the case of plane strain conditions no transversal deformation of the planar model is presenti.e. εz = 0 and consequently, for principal stress component σ3 transversal to the plane of planarmodel we have

σ3 = ν(σ1 + σ2) . (2.37)

After substitution of Eq. (2.37) into the relations of Eq. (2.34) and following adaptation in thecase of isotropic material (C1 = C2 = C0) we can write the expressions for the value of sum ofprincipal stresses

∆n = C0(1 + ν)(σ1 + σ2) . (2.38)

Fig. 2.17 illustrates the isopachic fringes around the crack tip recorded by double-exposure holo-gram. The phase change of the interference effect has originated from change of refractive indexof the PMMA transparent material.

All the experimental approaches mentioned above are based on the basic elasticity equationsfor plane stress/strain in planar models. As a matter of fact the interferometry on transparentplanar models can provide information only about the invariant of sum of principal stresses as itis described by Eq. (2.36) or by Eq. (2.38). However, in mechanics also the distribution of theindividual principal stress components often has to be important.

Photoelasticimetry based on temporary artificial birefringence appearing in optically isotropicbody forced by an external loading, belongs to the widely used methods in experimental stressand deformation analysis. The first authors dealing with holographic photoelasticimetry inter-preted the interference pattern as a superposition of independent sets of isochromatics – linesof equal differences – and isopachics – lines of equal principal tensions sums [12, 13]. It wasturned out, though, that such a simplistic understanding is not correct and the structure of inter-ference pattern is much more complex [14,15]. Further development of the method was focusedon isopachics and isochromatics differentiation aiming to obtain both of them simultaneouslybut independently. Several techniques were proposed by introducing additional optical elementssuch as rotator [16], depolarisation of information wavefront [17], etc. However, technical diffi-culties in practical realisation hamper their wider use to solve specific tasks.


n

R1 d' R2

RA

N1

Zd1S0

S2

S1

F N2 P T

Fig. 2.18. Scheme of holographic polariscope.

2.4.1 Polarisation holography

In polarisation holography the Jones formalism is a useful tool to describe the state of lightpolarisation. The Jones vector

S =(Sx

Sy

), (2.39)

where Sx and Sy are the components of the vector of light in the directions of axes x and y,is a compact notation of complex amplitude of monochromatic planar light wave. Influence ofoptical element on impinging polarized wave S is characterized by the Jones matrix

J =(j11 j12j21 j22

). (2.40)

The vector S′ of the resulting light is the product of the Jones matrix and vector of initial field

S′ = J · S . (2.41)

The light wave intensity is expressed as

I = S+ · S = S∗xSx + S∗ySy , (2.42)

where S+ means conjugate transpose, and a star denotes complex conjugate.Let us consider the holography interference polariscope scheme as in Fig. 2.18. N1 is an

element providing appropriate polarization state of the reference wave as well as the wave illu-minating objectM . Usually it is linear or circular polarizer, its properties can be written using theJones matrix JN1. Suppose to have linearly polarized monochromatic wave S0 with wavelengthλ and zero initial phase in front of N1. The Jones vector of the wave is

S0 = 2a(

10

), (2.43)


where 2a is the scalar aplitude of the wave. Let N1 be quarter-wave plate with azimuth 45◦, then

JN1 =1√2

(1 ii 1

). (2.44)

Light wave after passing N1

SN1 = JN1 · S0 =2a√

2

(1i

)(2.45)

is split on aplitude splitter Zd1 into two spatialy separated circularly polarized waves with equalscalar amplitude:

1. Reference wave S2 propagating to the plane of record F

S2 = B2

(1i

), B2 =

a√2

exp(ikn0RB) , (2.46)

where n0 is the index of refraction of surrounding enviroment, n0RB is the optical path,k = 2π/λ is the wavenumber;

2. The wave illuminating an object

S′M =

a√2

exp(ikn0R1)(

1i

). (2.47)

Optical properties of the object M in planar stress state can be characterized by the Jonesmatrix

JM = exp(iδ)(J11 J12

J21 J22

),

J11 = i cos 2ϕ sin δ + cos δ ,J12 = J21 = i sin 2ϕ sin δ ,J22 = −i cos 2ϕ sin δ + cos δ ,

(2.48)

where

δ = kn1 − n2

2h′, δ = k

n1 + n2

2h′ (2.49)

n1 and n2 are the principal indecies of refraction of anisothropical object with the thickness h′,ϕ is the direction of the principal index n1. Based on Maxwell-Neumann’s law for the planarstress state

n1 − n = C1σ1 + C2σ2, n2 − n = C1σ2 − C2σ1 , (2.50)

we have

δ = kC(σ1 − σ2)

2h′, δ = k

[D(σ1 + σ2)

2h′ + nh′

], (2.51)


C1, C2, C = C1−C2, D = C1 +C2 are the optical constants, n is the index of refraction of thenot driven (opticaly isotropic) object, σ1, σ2 are principal stresses, φ is the isoclines parameter.

The wave in the record plane F :

S1 = B1

(JF

1

JF2

), B1 =

a√2

exp[ikn0(RA − h′)] exp(iδ) ,

JF1 = J11 + iJ12 = i exp(i2ϕ) sin δ + cos δ ,

JF2 = j21 + iJ22 = exp(i2ϕ) sin δ + i cos δ .

(2.52)

Holographic record of isochromatics

In the considered layout, the superposition of the circularly polarized reference wave S2 andelliptically polarized information wave S2 is registered in on the recording medium:

S = S1 + S2 =(B1J

F1 +B2

B1JF2 + iB2

). (2.53)

Using reconstruction wave S3

S3 = B3

(1i

), B3 =

a√2

exp(ikn0RB) , (2.54)

with the scalar amplitude a, the wave Sv is restored according to the basic equations of holo-graphic imaging

Sv =a2a√

2exp[ikn0(RA − h′)] exp(iδ) cos δ

(1i

), (2.55)

which creates virtual image with the intensity

Iv = a4a2 cos2 δ (2.56)

carrying information about the relative birifrengence distribution or the function of differ-encies of principal stresses in the object respectively. The intensity minimas (cos2 δ = 0), i.e.isolines of the function

σ1 − σ2 =λ

Ch′m, m =

12,32,52, · · · (2.57)

create half-order isochromatics, the layout corresponds to the scheme of the classical circularpolariscope with bright view-field.

Double exposure record

In unloaded body of thickness h, n1 = n2 = n and the light field in the plane F is created bywaves

S01 = B0

1

(1i

), B0

1 =a0√2

exp(ik[n0RA + (n− n0)h]

), (2.58)

S02 = B0

2

(1i

), B0

2 =a0√2

exp(ikn0RB) . (2.59)


Light field S0 = S01 + S0

2 is recorded and S0v is then restored in recondtruction precess:

S0v =

a20a√2

exp(ik[n0RA + (n− n0)h]

)( 1i

)(2.60)

with the virtual image intensity I0v = a4

0a2 without any information about the object.

Then the information wave S1, Eq. (2.52), from the object with loading superponed on thereference S2, Eq. (2.46), is recorded using the unchanged optical setup. The reconstructed waveS = S0

v + Sv of such double exposed record has the intensity

I = a40a

2(1 + η2 cos2 δ + 2η cos δ cos δ) , (2.61)

where the following substitution are introduced

η = (a/a0)2 ,δ = k[n0d

′ + (n− n0)h]− δ ;(2.62)

expression for δ can be modified with respect to Eq. (2.49) and Hooke’s law for transverse de-formation of planar objects to the form

δ = kh

[ν

E

D

2(σ1 + σ2)2 −

(D

2− ν

E(n− n0)

)(σ1 + σ2)

], (2.63)

wherein the first term is usually neglected [13] and δ is considered as a sum of the principalstresses

δ =πh

λ

[2νE

(n− n0)−D

](σ1 + σ2) . (2.64)

The minima of cos δ for δ = sπ determines isofringes of the orders of s

σ1 + σ2 = Css ,

Cs =λ

h

[2νE

(n− n0)−D

]−1

,

s = ±12,±3

2,±5

2, · · · .

(2.65)

Real time analysis

Suppose the information wave recorded at the first exposure came from the unloaded object.The polarization state of the reconstructed wave S0

v, Eq. (2.60), is determined by polarizationof the reconstructing wave S3. The wavefront S1 from the object under load is incident on thishologram placed in its original position. Given that only parallel components of light vectors caninterfere, the intensity of the final field S0

v + S1 is

I = a40a

2(1 + ξ2 + 2ξ cos δ cos δ), ξ =a

a20a. (2.66)

Complement of the observation optical scheme by analyzer is proving to be an advanta-geous modification of holographic interferometric polariscope in real-time mode. In Fig. 2.18,


N2 denotes a quarter-wave plate characterized by Jones matrix JN2 = JN1. P is a polarizertransmitting the wave with oscillation parallel to S0. Then, light field behind the analyzer is

S =1√2

(0 0i 1

)(S0

v + S1) (2.67)

and the intensity

I = a40a

2(1 + ξ2 cos2 δ + 2ξ cos δ cos δ) . (2.68)

2.4.2 Interpretation of interference pattern

The basic layout of the holographic interference polariscope allows to obtain two types of inter-ferograms. Eq. (2.61) or Eq. (2.68) and Eq. (2.66), respectively, are charaterized by discontinousnature of the superimposed networks of interference fringes. Eq. (2.66) (originally derived forthe Mach-Zehnder classic interferometer [18]) in contrast to equations Eq. (2.61) and Eq. (2.68)does not contain the member cos2 δ, whose influence is significant:

1. Interference fringences according to Eq. (2.61) and Eq. (2.68) interpreted as isochromatesof half-number orders are always continuous lines, while interpreted as isopachics arediscontinuous.

2. Using Eq. (2.66) isochromates are continuous and isopachics are discontinuous, for theratio of spatial frequencies p > 1, on the other hand isochromates are discontinuous andisopachics are continuous for p < 1.

The interference pattern is rather a complicated function of spatial frequencies, cross-angleof superposed networks and amplitudes of light waves used in the recording and observation[14]. The useful technique in its interpretation shows to be the method of light field singularitiesenabling more precisely localisation of areas, on which light field intensity reaches zero values,i.e. I = 0. The condition is fulfilled, if

sin δ = 0 and α cos δ + cos δ = 0 (2.69)

(α ≡ η and α ≡ ξ), i.e. if

cos δ = 1 and cos δ = −1/α (2.70)

or

cos δ = −1 and cos δ = 1/α . (2.71)

The zero-intensity points lie on the isopachics of full line orders δ = s′π, s′ = 0,±1,±2, . . . ,while

1. for the interference pattern of type Eq. (2.66) they are present only when α = 1 and canbe found alternately at the intersections of full order isochromtas and isopachics.

2. for the pattern of type Eq. (2.61), Eq. (2.68) the singular points appear when

(a) for α = 1 in the places as in the case 1


Fig. 2.19. Interference fringes of both isochromatics and isopachics obtained by double-exposure hologra-phy on a planar transparent photoelastic object [19].

(b) for α > 1 doubled ponits I = 0 in the places cos δ = ±1/α, which are shifting forthe rising α along isopachic symetrically towards the visible half-order isochromates.

Complex nature of intensity distribution in the interference pattern is ilustrated in Fig. 2.19for simulated model of orthogonal networks of isochromats and isopachics superposition.Also the isofringes I = konst. for different values of α are plotted in the figure.

2.5 Visualisation of ultrasonic bending wave propagation

The ultrasonic testing of the concrete abounds in the problems and there are evidently possibili-ties, how the current diagnostic procedures can be improved. Complicated circumstances, whichare as a rule present in the case of heterogeneous and porous materials like concrete follow fromthe fact, that attenuation properties restrain using of higher frequencies – usually the frequencies150–200 kHz are considered as a limitation. Corresponding values of the ultrasonic wavelengthspropagating in concrete are going on in the range ∼ 2–20 cm i.e. these lengths are comparablewith geometrical dimensions of the specimens tested. It gives rise to problems with the sensingof waves arriving from the reflected free boundaries of the specimen and with the following sep-aration of longitudinal and shearing wavefronts. On the other hand, the considerable dependencyof the attenuation bears information about the main mechanical parameters of the concrete andits deterioration, and thus makes its evaluation possible. As it is turned out, the key factor atthe precise sensing of both the wave velocity and the attenuation is the ability of transducers forright acquisition of the time dependency of mechanical vibration on the concrete surface inducedby the ultrasonic wave. Piezoelectric accelerometers or simple US probes conveniently used forsuch a purpose take information by contact means and its transient characteristic depends on thetight fixation of the transducer on the surface.

Holographic interferometry using pulsed laser has long been applied for the visualisation ofstructural vibrations. The pulse width or exposure time of the appropriate lasers is usually sometens of nanoseconds, thus even the dynamic deformation of the surface stress wave propagationcan be recorded by hologram without any problems with blurring. Holloway [20] and Apra-hamian et al. [21] demonstrate the use of the method for observation of a mechanical excitement


propagation in plates.Using opportunities of the pulsed ruby laser the flexural ultrasonic waves propagation has

been studied in the concrete plates specimens of the widths 33 mm, 55 mm and 120 mm. Thediameter of the freely supported plates 650 mm, as well as the duration of the exciting impactwere chosen from the condition to avoid the interference of the waves reflected from the freeboundaries of plates at the moment of measurement. The diameter of the freely supported plates,as well duration of the exciting impact were chosen 650 mm and 20–30 µs, respectively. By sucha means the propagating circular bending waves create a few periods of waves on the concretesurface before their reflection from the free boundary.

The impact of projectile from the air gun the mechanical impulse excited the vibrations inultrasonic region of frequencies. The impact speed of the lead projectile was measured photo-electrically. The dispersion of individual fires was ±5 m s−1 at the impact speed 150 m s−1. Thetime traces of the impact force were recorded by two sensing devices. One of them was standardpiezoelectric accelerometer fixed precisely in the place of projectile impact on the opposite sideof the plate. By double integrating of the electric signal from the accelerometer the time historyof its displacement was deduced and consequently, the dynamic moving of concrete surface atthe fixation point was evaluated.

In order to measure the dynamic impact force directly, miniature dynamometer was devel-oped. It is based on optical diffraction principle [22], where the light of laser beam is focusedonto the narrow slit in the centre of elliptically shaped steel ring (the outer diameter Ø = 10 mm).The light after passing through the slit is diffracted and collected onto the effective area of pho-todiode. Deformation of the steel ring owing to acting force changes the width of the slit andsubsequently the amount of the light passed.

The requirement to develop the dynamic force transducer with higher value of its natural fre-quency is followed by the necessity of appropriate calibration in absolute values of the force. Themost simple way how to do the calibration seems to be static calibration force vs. output electricsignal. However, in such a case the transducer cannot be used reliably for measurement of forceimpulses with frequencies comparable with that of resonant (natural) frequency of transducerdue to the dynamic nonlinear effects at this region. Despite its miniature dimensions the naturalfrequency of the sensor was about 100 kHz, that is why we have developed the dynamic methodof calibration based on Newton’s second law. Since force duration is an impulse, we write theknown relationship

mv = −∫ T

0

Pdt , (2.72)

wherem and v are the mass and the impact velocity of the projectile, respectively, and P is actingdynamic force during the time T . As it follows from Eq. (2.72) to determine the absolute valuesof time varying dynamic force also its time history is needed. This was recorded by oscilloscopeand proved to be nearly half-sine shaped (see Fig. 2.20).

Half-sine force peak as it is assumed by the Hertz’s theory of impact was affirmed, in spiteof the fact, that the projectile made of lead was deformed at the impact plastically

P (t) ={Pmax sin(πt/T ) , 0 ≤ t ≤ T ,0 , T ≤ t .

(2.73)


Fig. 2.20. Oscilloscopic trace of the dynamic force generated by projectile impact on the plate surface.Time base 10 µs/div, dynamic force scale 293N/div.

The maximum magnitude level of the force Pmax for the half-sine shape mentioned, can becalculated by expression

Pmax =πmv

2T. (2.74)

Such a dynamic calibration of the tranducer is sufficiently precise, however, due to the thermaleffect the part of kinetic energy can influence the results. To remove this factor, this thermalamount of energy was experimentally determined by the measurement of energy transferred toheavy weight pendulum oscillations.

The frequency content of the energy applied to the structure is a function of the stifness ofthe contacting surfaces, the mass of the impacted projectile and the velocity of its impact. Thestiffness of the contacting surfaces and, to some measure, the shape of the projectile affectsthe time history of the force pulse, which in turn determines the frequency content of excitingvibrations. As it is not feasible to change the stiffness of the specimen, the frequency contentmay be controlled by varying the impact velocity and/or by a material of projectile. The actualfrequency content of the exciting pulses was assessed by Fourier spectral analysis of the timetraces recorded. The time width of such mechanical shocks was about 20–25 µs (see Fig. 2.20).This leads to the exciting of the ultrasonic frequencies up to 100 kHz, where the spectrum comesnear to zero. A disadvantage to note here is that the force spectrum of an impact excitationcannot be band-limited at lower frequencies, which are highly expressive. Nevertheless, thebroad-band spectrum includes all the frequency components (up to 100 kHz), so that the spectraldependencies, such as wavelengths and dispersions, can be acquired.

Double exposure holograms were recorded by means of two monopulses of ruby laser atthe wavelength of 694 nm. At the first exposure the hologram of concrete plate surface beforedeformation was recorded and then the second exposure after the time interval 5 µs to 120 µsafter beginning of the impact the second hologram was recorded. The different double-exposureholograms were recorded for variety of time moments after starting of impact. Interferometri-cal pattern recorded on holograms represents different position of the running wavefront fromthe impact point. Geometry of optical scheme was arranged with incident angle of illuminat-ing light nearly perpendicular to the object surface observed and the object was also abserved


Fig. 2.21. Sequence of holographic interferograms of the surface dynamic deformation on the concretesurface – 5 µs after load beginning, 25 µs, 50 µs, 65 µs and 125 µs.

through hologram nearly perpendicularly to the plate surface. Then, the interference fringes canbe interpreted as curves of equal amplitudes of plate deflections

w =Nλ

2 cosϕ, (2.75)

where N is interference order, λ is wavelength and ϕ is angle of incidence or observation. Atthis layout the fringe value used for the evaluation of interferograms was 3.87× 10−4 mm.

In order to synchronize the time of second exposure exactly at the moment of instantaneouswavefront position arbitrarily chosen, the simple electro-mechanical sensor was developed. Twopieces of aluminium foil were placed one to another with a narrow interspace and they were con-nected in electronic circuit. At the moment of the foils penetrating by flying projectile the shortconnection created basic triggering impulse. The sychronizing of the laser flash is complicatedby the fact, the moment of arriving of the input signal to the laser controller unit to pump thelaser ruby crystal and to break the passive Q-switch. Therefore, the precise positioning (in ourcase about 100 mm) of the sensor in front of the concrete surface was carried out tentatively.By small shifting of the braking sensor position this distance was corrected and the moment ofinstantaneous dynamic deformation of recording at the running wavefront was chosen for each ofthe case. The time interval between the starting of impact loading and the moment of the secondlaser light pulse was measured comparing the time position of signal transmitting from piezo-electric transducer fixed in the centre of the back side of concrete plate and from the photodiodesignal of laser light flash.

Holographic interferometry can give the large field view on the instantaneous dynamic de-formation of the surface. However, the detailed knowledge of the time dependency of this de-formation is of fundamental importance. On that account, we have used also the piezoelectrictransducer to confront its data with that obtained holographically.

To illustrate the possibilities of holographic interferometry, the time sequence of the holo-graphic records of flexural wave propagation on the concrete surface is shown in Fig. 2.21. Thesewere taken at different time moments after starting of loading. Note that the time history of dy-namic deformation was folded from different events of the loading series.

In order to compare the real situation with that of theoretically predicted we have calculatedthe instantaneous surface deformation state for several time moments. These calculations werebased on the analytical expressions obtained in [23]. The expressions take into account linear


Fig. 2.22. Distortions of ultrasonic wavefront propagating on a plate surface.

50 100 250 3500 150 200 300

Distance from the impact point mm

Our

-of-

plan

e de

form

atio

n w

µm

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

50 µs

MeasurementAnalytical calculation

Fig. 2.23. Comparison of analytically vs. experimentally obtained instantaneous deformation of the con-crete surface around the point of impact.

behaviour of the concrete as a homogeneous material and also approximation of dynamic loadingpulse is in agreement with the Hertzian theory.

One of the chosen time points of instantaneous dynamic deformation is plotted in Fig 2.22.The solid curve is the analytical prediction of surface deflections and crosses indicates the

deflection values read as interference orders from holographic interferogram. As seen, there is agood coincidence of calculated surface shapes with that observed by holography. Concerning thequantitative values of deflections, we have to note, that the dynamic values of Young’s modulusused in calculations were determined from ultrasonic measurements of concrete material, andalso the real values of the loading force at the impact were obtained from the data from diffractiondynamometer.

Besides the deformation calculations using the simplified analytical expressions the numeri-cal computation on the basis of Kirchhoff’s theory of plates were carried out, too. One result ofsuch a calculation of the dynamic instantaneous surface shape is presented in graph of Fig. 2.23.Here the circular plate with thickness of 55 mm was computed at the time moment of 30 µs after


the initiation of impulse loading. From the graph there is a clearly visible dispersion phenomenondue to relatively small thickness of the plate in comparison with stress wave wavelengths inducedby mechanical impact. Theoretically, the infinite frequencies have to propagate with the infinitevelocities of these frequencies. However in real situation, the highest frequencies (in the casemore than 100 kHz) are quickly attenuated, that is why these frequencies are not visible. Nev-ertheless, the edge of ultrasonic wavefront from holographic records is clearly contoured. Wellobservable are also distortions of propagating wavefront caused by subsurface inhomogenities ofthe heterogeneous concrete material (see Fig. 2.22).

2.6 Interferometrical evaluation of the temperature distribution inside transparency

Dealing with the clearly transparent objects through which the light is passing, the classicallaser based wide field interferometry can often be effective. Besides the standard testing of op-tical components, namely the transparent elements are proper examples where these approachesclaim attention. In transparency, not only the shape or deformation changes are often relevant,but also the refractive index variations or refractive index whole field distribution are interesting.As known, the refractive index of liquids, gases but also transparent solids show the tempera-ture dependency to such an extent that the temperature changes in units of degrees, or even partof a degree can be sucsessfully detectable interferometrically through path difference detection.Temperature derivative of the index of refraction is a characteristic value of the material, hence,after the appropriate calibration, an interferometrical mapping can provide also temperature fielddistribution inside the object even on quantitative basis. Besides this, the refractive index depen-dency is conveniently used for detection of pressure induced variations of the refractive indexfield. Mostly, such a measurement does not need a large field of observation, then the inter-ferometrical scheme can be assembled by small sized optical elements, which can be precisedenough and they are not too expensive. Despite the specific problem of the transparent objectshape inaccuracy, when rather a dense pattern of interference fringes is observed in initial state,can be overcome as well. Then the temperature field in solid transparency is evaluated by mutualcomparison of the interference patterns before and after the heating of the model.

Such an experimental interferometrical measurement has been performed to evaluate temper-ature distribution inside the second harmonic generation (SHG) KTP crystal. As it was approvedpreviously, the temperature change induced by multimode exciting beam of λ = 1064 nm isnonmeasurable when no second harmonic generation occurs. To overcome the current problemas well as lack of pumping, Raman shifted 1178 nm Yt:fiber laser and to have the possibilityto elaborate the detection and evaluation procedure, heating of the SHG crystal by CW argonlaser was used. The argon laser used emits basic spectral lines at λ = 514 nm and 488 nm withthe output power up to 6 W. The absorption coefficients of KTP crystal material at this spectralrange is approximately one order higher compared to the absorption at λ = 1064 nm. Moreover,the CW argon laser emits nearly TEM00 mode, thus the simulated exciting beam can be concen-trated inside crystal into the focus with the waist of 20–40 µm in diameter. Then, the conditionsof crystal illumination and consequently the heat generation are presumably not so far from thoseat second harmonic generation.

The crucial problem of the temperature distribution assessment is the limited sensitivity ofinterferometric method. The primary value to be measured is the index of refraction but its tem-perature induced changes are very small, currently at parts of degree change. Another limitation


is the fact, that the whole area of the SHG crystal has to be mapped. That is why photoelectricsensing based on high resolved photodiode cannot be used and also observation by CCD hasan insufficient light intensity resolution. Furthemore, when detecting the path differences muchsmaller than the wavelength of light, also the overall concept of interferometer setup has to beregarded. At such sensitive measurements the fluctuating ambient environmental conditions arean additional factor.

All of these factors lead to the proposal to install Fabry-Perot interferometry based opticalscheme where the differential comparison of both the front and backside of the crystal are re-alized. In such a setup only the variations of refractive index inside crystal are detected, othercircumambient influences are compensated and do not affect the resulting fringe pattern.

When a coherent beam of light is incident perpendicularly or nearly perpendicularly upon theSHG crystal sidewalls, the resulting pattern of interferency between both the reflected wavefrontsis created. The total field E at a point on observation screen (or CCD chip) is the sum of the twowaves superimposed on each other and results in the following

E = E1 + E2 = a[e−ikr + e−ik(r+h)

]eiωt . (2.76)

In the above equation A is the nominal amplitude of the laser light, k is the wave number, r and(r + h) are the optical paths after reflection off the sidewalls, ω is the angular frequency and tis the time. Notice that multiplying h by the wave number k, is a representative of the fringepattern phase shift

ϕ =2πλh . (2.77)

Rearranging Eq. (2.76) in terms of the phase shift results in

E = ae−i(kr−ωt)(1 + e−iϕ) . (2.78)

The term in parenthesis can be rewritten as 2 cos(ϕ/2)e−iϕ/2 and multiplying the equation by itscomplex conjugate yields the intensity distribution as a function of the fringe pattern phase

I = |E|2 = 4a2 cos2(ϕ/2)

or

I = 2a2(1 + cosϕ) . (2.79)

When the very small index of refraction variations has to be evaluated, the necessity of precisemeasurement of subfringe shifting of the interference pattern is vital part of evaluation. Up to day,such a data evaluation has been performed usually on subjective basis and then the error reached1/10 to 1/20 of the width of fringe contour. In principle, it is possible to detect intensity changeswith much higher sensitivity by a photodiode based photoelectric sensing . Unfortunately, suchmeasurement is the only point-like method and mapping of whole the area of crystal will needsome kind of scanning. We have tested also such an approach as described later in this section.

In the case of demand for the whole field of observation, image processing must be applied inorder to compare both the interference patterns, before and after heating of the crystal by excitingAr laser.


x

B 0

B ( x )

x 0 x S 1

Fig. 2.24. Double-beam intensity profile of the interference fringes.

By a simple mathematical derivation based on the previous expressions, the function of in-tensity distribution in the field of view on plane of interference fringes localization can be written(see also Fig. 2.24)

I(x) = 2I0

(1 + γ cos

[2πd1

(x− x0)])

, (2.80)

where x0 is the coordinate of the reference interference fringe, I0 is the intensity of one of inter-fering wavefronts, d1 is the width of the interference fringe, and γ is the constant representingfringe contrast. Eq. (2.80) describes the system of parallel interference fringes in the directionperpendicular to the fringes and can be used also to express the fringe pattern after its shiftingdue to the temperature index of refraction change

I(x′) = 2I0

(1 + γ cos

[2πd1

(x+ x′ − x0)])

, (2.81)

where x′ is the shift of interference fringe.As seen, comparison of both the equations Eq. (2.80) and Eq. (2.81) gives us the possibil-

ity to determine the path difference induced by temperature increase through the evaluation ofseparated fringe maxima or minima shift x′. As the change of path difference between twoneighbouring fringes is equal to wavelength of light used, the non-dimensional fringe patternshift relative to its initial position gives the searched change in path difference

∆l =x′

d1λ . (2.82)

Consequently, the non-dimensional fringe shift can be measured by evaluating the initial locationof a fringe minimum or maximum and tracking how it moves relatively to its primary positionand dividing it by the period of one fringe cycle.

Taking into account the double passing of the SHG crystal thickness

∆l = 2h0(n− n0) + (h− h0)n0 , (2.83)

where n0 is the initial index of refraction inside the crystal in the direction of light propagation,h0 is the thickness of SHG crystal, n is the index of refraction inside the crystal after its heating,and h is the thickness of the crystal after heating.


With regard to the real values of index of refraction vs. temperature and on the other hand thecoefficient of thermal expansion of SHG crystal material, the second term of Eq. (2.83) can beomitted, its quantitative value is as much as one order smaller than the path difference caused byindex of refraction change. Finally, using the known parameter of temperature derivative ∂n/∂T ,where T is temperature, the required value of the inside temperature increase can be determined.

In real conditions of the SHG crystal measurement both the expressions Eq. (2.80) andEq. (2.81) have to be modified. As a rule, the prism of the crystal is ground with non-preciseplanparallelity of its surfaces, thus the interference fringes of initial patterns are deformed. Pro-vided that the fringe of equi-phase points (e.g. fringe of minima or maxima) can be described bya simple function

y = y(x) , (2.84)

the equation of the pattern intensity distribution is as follows

I(x, y) = 2I0

[1 + γ cos

2πd2

((x− x0) sinα+ y sinα

)], (2.85)

where α is the angle between the tangent to fringe curve and x axis and d2 is the width ofinterference fringe in the direction perpendicular to the fringe curve at the point under consider-ation. Using the Eq. (2.82) also at such a pattern the path difference temperature change can bedetermined.

All the above mentioned expressions were derived assuming the double-beam interferenceconditions. However, the SHG crystal is a nearly plane-parallel plate and in fact Fabry-Perotmulti-beam interference is registered in the scheme with both the front and the backside reflec-tions. When a monochromatic light is directed normally to the transparent plate having refractiveindex n0 and the thickness h0, the reflectance is given by well known Fresnel expression

RF =R1 +R2 − 2

√R1R2 cosϕ

1 +R1R2 − 2√R1R2 cosϕ

, (2.86)

where R1 and R2 are the reflection coefficients for the front surface and the back one, and

ϕ =4πλn0h0 (2.87)

is the optical path difference of both the interfering beams. For such a multi-beam interferencethe intensity distribution of the fringe pattern is as follows

I(x) = I0(x)F sin2 ϕ/2

1 + F sin2 ϕ/2, (2.88)

where

F =4R

(1−R)2; ϕ =

2πλl1 cos θ (2.89)

and R = R1 = R2 is the coefficient of reflection of the SHG crystal surfaces, θ is the anglebetween direction of wavefront propagation and the normal to the surface, and

l1 = 2n0h0 (2.90)


1 2 3 4 50 , 0

0 , 2

0 , 4

0 , 6

0 , 8

1 , 0

Inten

sity a.

u.

x m m

Fig. 2.25. Shape of the multi-beam interference fringes.

is the optical path.The illustrative example of the obtained fringe pattern distribution is shown in Fig. 2.25 where

the deviation of wavefront curve is visible from the pure harmonic development of Eq. (2.80).Nevertheless, the surfaces reflectance of the measured object of crystal is in the range of somepercent (R ≈ 0.05), then for the small value of F the shift of the fringe pattern is nearly thesame as described by Eq. (2.82).

As it follows from the results obtained (see the section below), the evaluation proceduredescribed can be successfully applied to map the temperature distribution throughout the wholearea of the SHG crystal. In practical realization the limit of ultimate sensitivity for temperaturechange evaluation can be given using realistic assessment for x′/S1 from 1/20 to 1/40, or 1/20λto 1/40λ (see Eq. (2.82)). With the actual temperature derivative ∂n/∂T parameter of the KTPcrystal it leads to the experimental uncertainty of temperature defining of ±0.1 to ±1.0 K.

In order to overcome this ultimate value, the point like measurement has to be used with thedetecting of fringe patter intensity variations by photodiode detector. Again, based on Eq. (2.79)for small intensity changes ∆l� λ we obtain by differentiation following expression

∆I(x) = a2 2π∆lλ

sinϕ . (2.91)

From that equation the change in paths differences is written

∆l(x) =∆I(x)λ

2πa2 sin(2πl/λ). (2.92)

The expression of Eq. (2.91) indicates the best sensitivity of intensity sensing at the positions,where

ϕm =π

m+ 1/2, (2.93)

that is in the central positions between neighbouring interference maxima and minima. There-fore, the active area or slit diaphragm of the detector has to be placed at this position.


(a) (b)

Fig. 2.26. Interference fringes of equal thickness of the KTP crystal (a) before laser heating, (b) after laserheating.

In the case of multi-beam interference starting from the Eq. (2.88) the expression for intensityvariations due to small path difference changes can be derived

∆I(x) =π∆lλ

F sinϕ/2(1 + F sin2 ϕ/2)2

(2.94)

and then for change in optical paths differences

∆l(x) =∆I(x)λ

π

(1 + F sin2 ϕ/2)2

F sinϕ/2. (2.95)

Fabry-Perot interferometrical setup with the observation of interference pattern by digitalcamera was arranged. The sensitivity of interference pattern reading was sufficient to detect theheating of the SHG crystal induced by focused Ar laser beam at the power of 2 W and 5 W.As an illustration in Fig. 2.26 there are the interference patterns of the crystal around the focusof the Ar exciting beam. The figures present the initial temperature state and the heated state.The crystal is situated on the brass base with good heat removal. In Figures the brass base isin the bottom part, vertical dimension means 5 mm of the crystal width. Fig. 2.27 shows thecomparison of light intensity distribution for two states of the crystal temperature – the initialand the steady state after its stabilising. The transient process of heating lasts about 60 seconds.Duration of the reversible process of the crystal cooling was approximately the same.

Next Fig. 2.28 presents the experimentally determined temperature distribution inside mea-sured KTP crystal and its comparison with the results of numerical simulation. As it can besimply deduced, the nature of curves is the same although the quantitative values of temperaturecan be strongly influenced by a not very precise knowledge of the parameters ∂n/∂T as well asby basic thermomechanical constants for numerical input data.

The interference patterns observed were also analyzed by photodiode detector. The activearea of the photodiode was screened by slit diaphragm whereby the slit opening was properlyoriented and positioned at the best point between neibouring fringes. The crystal area was thenmapped point by point in ten chosen positions.


0

20

40

60

80

100

Oven

Air

ColdHeated

Inte

nsi

ty a

.u.

x mm-2 -1 0 1 2 3-3

Fig. 2.27. Shifting of the light intensity interference pattern after heating the crystal.

-2 -1 0 1 2 3300

301

302

303

Air

Tem

pera

ture

K

xmm-3

Bra

ss s

upport

Fig. 2.28. Comparison of the experimentally obtained temperatures with the numerical simulation.

2.7 Membrane thickness mapping by Fizeau interferometry

The measurement of membrane thickness variation is one of the important points of its mechan-ical characterization. Since the membrane can be regarded as an element similar to that of thinplanparallel plate, the Fizeau interferometric scheme makes an advantage of a rather simple ex-perimental realization. As known, in the near IR region silicon material is transparent, but at thethickness of no more than several micrometers, the silicon transparence even for visible region isenough to pass the light twice through the thickness. The coherent light reflected from both thefront and the back membrane surfaces interferes and creates the interference fringes. The lightintensity in the interference field of two beams is expressed by the known equation

I = I1 + I2 + 2√I1I2 cos

(4πλnh− π

), (2.96)

where I1, I2 are the intensities of the beams reflected from the front and the back sides, respec-tively, λ is the light wavelength, n is the index of refraction of the membrane material and h is


h

n

Light source

Camera

Fig. 2.29. Fizeau interferometrical scheme for membrane equal thickness mapping [24].

the membrane thickness.The coefficients of light reflection at silicon-air interfaces are favourable due to high value of

silicon index of refraction, and for the red light they are almost optimal to secure approximatelyequal amounts of light reflected backward from both the membrane sides. In the case of weakabsorption contrast of the interference fringes is defined by the following expression

η =2(1−R)e−αh

1 + [(1−R)e−αh]2, (2.97)

where R is the reflectivity of the membrane surfaces at perpendicular illumination

R =(n− 1n+ 1

)2

(2.98)

and α = 4πκ/λ is absorption coefficient, κ is the imaginary part of index of refraction.Inserting into Eq. (2.98) the actual value of the index of refraction (n = 3.89) for silicon

material at λ = 632.8 nm (He-Ne laser) an excellent contrast of interference pattern is obtainedwithout any absorption: η = 91 %. The light absorption in silicon slightly reduces this value toη > 83 % (see Eq. 2.97) but still the contrast remains very good. When green laser (λ = 532 nm)was used, the contrast showed significant decrease to the value of η = 56 % caused mainly bystronger absorption.

Besides the good stability, the Fizeau interferometrical scheme realization provides also dis-tinct fringe value – the constant ch which characterises the thickness change between two neigh-bouring fringes.

∆h = Nλ

2n= Nch . (2.99)

Regarding the high value of silicon index of refraction and also the double passing of the lightthrough the membrane, the measuring sensitivity – the fringe value is as high as ch = 81 nm.Moreover, the sensitivity can be increased substantially by numerical image processing usingfunctional interpolation between neighbouring fringes. If necessary, by such a way the thicknesschanges are detected with the resolving power of a few nanometers.

To observe the interference fringes, a simple optical scheme was arranged (see Fig. 2.29). Inthe experimental setup, He-Ne (632.8 nm/40 mW) laser was used and the interference patterns


Fig. 2.30. Interference fringes of the membrane thickness variations.

were recorded alternatively by photographic or CCD camera. In the setup, a large 6-inch diameterobjective with focal distance f = 1 000mm was used. By such a means, the whole area of themembranes up to 150 mm in diameter was observed.

As an illustrative example, the map of thickness variations is shown in Fig. 2.30. Goodcontrast and lateral resolution of the pattern enables us to see also the details at the membraneboundaries after optical magnification. As the interference fringes are visible also on a part ofcarrier SOI wafer silicon ring, potential defects of the membrane boundary can be inspected.

Speckle interferometry 143

3 Speckle interferometry

3.1 Optical system of holographic/speckle interferometer

Simplified scheme of basic optical setup of the holographic interferometer is shown in Fig. 3.1.Spherical lens with focal distance f projects the image of the object surface to an image planez = z2 , where a hologram is created by interference with off-axis reference wave. A diffusellyscattered surface of the object is deformed by loading or moving in the interval between the firstand the second exposures of double-exposure hologram.

The double-exposure image-plane hologram is registered as a result of interferency of thereference wave a(x0, y0) with the waves in the image plane corresponding to the both states ofthe observed object. The surface of object is illuminated by a plane wave with complex amplitude

aI(x, z) = aI exp[i(k(x cosαx + z cosαz) + ϕ)] , (3.1)

where aI is the amplitude of light field, ϕ is the phase of the wave at the origin of coordinatesystem and cosαx, cosαz are the direction cosines of the wave along the light propagation,k = 2π/λ (where λ is the wavelength) is the wave number. Complex amplitude immediatelyafter the reflection off the object is as follows

a0(x0, y0) = r (x0, y0) exp[ik(x0 cosαx + z0 cosαz)] , (3.2)

where r (x0, y0) is the amplitude reflectance of the object surface and the initial phase φ at thepoint P equals zero.

As we consider the linear optical system, field amplitude in the image plane z = z2 can beexpressed by superposition integral

a2(x2, y2) =∫∫ ∞

−∞a0(x0, y0)h(x0, y0, x2, y2)dx0 dy0 , (3.3)

where h(x0, y0, x2, y2) is the impulse response of the system, i.e. amplitude formed in the planez = z2 by a point-like source located at the point P (x0, y0).

Fig. 3.1. Simplified scheme of basic optical set up of the holographic interferometer.


To acquire the point response of the system, in the object plane we have chosen a point-likesource described as δ-function with coordinates (x0, y0,−w). Then, after passing the wavefrontthrough the diaphragm with aperture function P1(x1, y1) and the lens we obtain the well knownimpulse response in the image plane

a2(x2, y2, x0, y0) =eikq

iλq

∫∫ ∞

−∞a+1 (x1, y1)

× exp[ik

(x2 − x1)2 + (y2 − y1)2

2q

]dx1 dy1 ,

(3.4)

where now we have not considered the diaphragm in focal plane. After proper substitutions wehave

a2(x2, y2, x0, y0) =eik(p′+q)

λ2p′qexp

[ik(x2

0 + y20

2p′+x2

2 + y22

2q

)]∫∫ ∞

−∞P1(x1, y1)

× exp[ik2

(1p′

+1q− 1f

)(x2

1 + y21

)]× exp

[−2πi

(x0

p′+x2

q

)x1 − 2πi

(y0p′

+y2q

)y1

]dx1 dy1 .

(3.5)

Considering the very small values of the out-of-plane displacements, which is allways the case,at the first approximation when w � p we can use approximation

1p′

=1

p+ w≈ 1p− w

p2, (3.6)

With the following substitutions

ξ2 =1λq

[x2 +Mx0

(1− w

p

)]; η2 =

1λq

[y2 +My0

(1− w

p

)], (3.7)

the expression Eq. (3.5) becomes

a2(x2, y2, x0, y0) = exp[ik(w + p+ q +

x20 + y2

0

2(p+ w)+x2

2 + y22

2q

)]×∫∫ ∞

−∞P1(x1, y1) exp

[ik2

(− w

p2+

1p

+1q− 1f

)(x2

1 + y21

)]×e−i2π(ξ1x1+η1y1) dx1 dy1 ,

(3.8)

a2(x2, y2, x0, y0) =

J1(2π%2b1)%2

exp[ik(w + p+ q +

x20 + y2

0

2(p+ w)+x2

2 + y22

2q

)],

(3.9)

where J1(2π%2b1) is the Bessel function of the first kind and first order and %2 is the polar radiusin coordinate system ξ2, η2. The Bessel function forms spherically symmetrical area with the


centre in (x2, y2) at ξ2 = η2 = 0

x2 = −Mx0

(1− w

p

); y2 = −My0

(1− w

p

). (3.10)

In addition to the amplitude, the holographic record registeres also information about phase dis-tribution of the light field. Therefore, the phase terms in the expressions cannot be ignored asit is usually done. It can be seen from the expression Eq (3.10) that the phase distribution ofthe field is also a function of coordinates of the surface point, which is the second phase factorin Eq (3.9). This effect complicates the evaluation of obtained double-exposure interferograms.Taking into account that the quadratic term represents the parabolic approximation of the spher-ical surface with the radius p′ centered at the center of the lens, small objects do not have toconsider it. It implies that uncertainty in the phase differences can be neglected when the objectis several times smaller than the lens, as it is in the order of tenths of a percent. For objects withdimensions comparable with that of lens diameter it can be avoided by the introduction of thebinary spatial filter-diaphragm with a small hole in the focal plane which suppresses the phasefactor. In addition the filter, as we show further, allows to change the cross-correlation conditionsbetween records at the first and the second exposures.

Let the diaphragm with the diameter of circular opening bf be shifted off the optical axis toa distance df (Fig. 3.1). Then the complex amplitude at the image plane is as follows

a′f (xf , yf ) =eik(1−f)

iλ(q − f)

∫∫ ∞

−∞Pf (xf , yf )a−f (xf , yf )×

exp[ik

(x2 − xf )2 + (y2 − yf )2

2(q − f)

]dxf dyf ,

(3.11)

where Pf (xf , yf ) is aperture function of the binary filter. After substituting the complex ampli-tude immediatelly after the lens and the amplitude in front of the filter into Eq. (3.11) and nextrearranging of the terms we get

a′2(x2, y2, x0, y0) = exp[ik(p′ + q +

x22 + y2

2

2(q − f)

)]∫∫ ∞

−∞Pf (xf , yf )

× exp

[ik

(x2

f + y2f

2f+x2

0 + y20

2p′

)]∫∫ ∞

−∞P1(x1, y1)exp

(ikx2

1 + y21

2f

)× exp

[−2πi

(xf

λf+x0

λp′

)x1 − 2πi

(yf

λf+

y0λp′

)y1

]dx1dy1

× exp

[ik

(x2

f + y2f

2(q − f)− x2xf + y2yf

q − f

)]dxf dyf ,

(3.12)

where we have omitted constant factors. Considering that r1 � rf , the aperture functionP1(x1, y1) can be set equal to one through the whole plane (x1, y1), and the impulse responseof the system is determined in a considerable degree by the aperture Pf (xf , yf ). Then the innerintegral represents a Fourier transform of exp

[iπ(x2

2 + y22)/λp′

], and its solution in domain of

spatial frequencies ξf , ηf is [26]

Yf = iλp′e−iπλp′(ξ2f +η2

f ) , (3.13)


where

ξf =1λ

(xf

f+x0

p′

); ηf =

1λ

(yf

f+y0p′

). (3.14)

After the substitution into Eq. (3.12) and following adjustment, the quadratic termexp

[iπ(x2

0 + y20)/λp′

]vanishes and for a′2(x2, y2, x0, y0) we get

a′2(x2, y2,x0, y0) = eik(p′+q) exp[2

ikq − f

(x22 + y2

2)]

×∫∫ ∞

−∞Pf (xf , yf ) exp

[ik2f

(1− p

q− p

f

)(x2

2 + y22

)]× exp

[−2πiλ

(x2

q − f+x0

f

)xf −

2πiλ

(y2

q − f+y0f

)yf

]dxf dyf .

(3.15)

By applying the approximation of Eq. (3.6), the integral in Eq. (3.15) is treated again usingFourier transform in domain of spatial frequencies ξ′2, η

′2:

ξ′2 =1λ

(x2

q − f+x0

f

); η′2 =

1λ

(y2

q − f+y0f

). (3.16)

Due to the circular symmetry of Pf (xf , yf ) let us rewrite the integral Y ′2 from Eq (3.15) in

polar coordinates

rf =[(xf − df )2 + y2

f

]1/2

,

θf = tan−1

(yf

xf − df

),

%′2 =(ξ′22 + η′22

)1/2,

ϕ′2 = tan−1

(η′2ξ′2

).

(3.17)

After substitution and simple modification we have

Y ′2 = e−i2πdf ξ′

2

∫ bf

0

rf drf∫ 2π

0

exp [−i2πrf%′2 cos(θf − ϕ′2)] dθf , (3.18)

the solution of which is analogous to Eq. (3.9) function J1(2π%′2bf )/%′2. The complex amplitudetakes the form

a′(x2, y2, x0, y0) = eik(p′+q) exp[2

ikq − f

(x22 + y2

2)]

× exp[−ikdf

(x2

q − f+x0

f

)]J1(2π%′2bf )

%′2.

(3.19)

However, the aperture presence in the focal plane of the lens in the reconstruction schememeans certain practical disadvantages. Small opening considerably extends exposure times and,


especially, this configuration does not allow to record simultaneusly displacement components inthe directions of the coordinate axes x0, y0. Therefore, the process of spatial frequencies filter-ing was postponed to the reconstruction phase. Thus, double-exposure holograms are recordedwithout any aperture in the focal plane. This approach offers also other benefits.

For linearly recorded double-exposure amplitude hologram, the relationship between ampli-tude transmissivity T and exposure is expressed as follows

T = T0 + β t{2a2

R + (a212 + a2

22) + [a21(x2, y2) + a22(x2, y2)] a∗z(x, y)+ [a∗21(x2, y2) + a∗22(x2, y2)] aR(x, y)} ,

(3.20)

where a21(x2, y2), a22(x2, y2) are complex amplitudes of the object waves in the first and thesecond exposure respectively, and aR(x, y) is the amplitude of the reference wave. The valuesT and β are characteristic constants of the photo-material and t is exposure time. In the recon-struction scheme, the reconstruction wave is chosen to be complex conjugate of the recordingwave. Illuminating hologram by such wave, complex amplitude of the −1st diffraction order inreconstructed wavefront immediately behind the hologram will be the second term of Eq. (3.20)in brackets

β t a2R[a∗21(x2, y2) + a∗22(x2, y2)] . (3.21)

It is desirable to carry out the reconstruction process in the same optical system as for thehologram recording. Photographically processed hologram should be situated nearly in the sameposition (oriented with emulsion to the lens) in the photo-plate holder. Afterwards it is illumi-nated by a plane wave spreading in the reverse direction as the reference wave during recording.Thus, the complex amplitude of the reconstruction wave would be a∗R(x, y). The use of planarwavefronts in the recording and reconstruction process simply removes problems that would ariseby distortion of wavefronts passing through the thick glass photo-plates as well as aberrations re-sulting from the fact that the photo-emulsion always creates both the amplitude and the phaserecord. By such a way we ensure that the waves generated in −1st order of diffraction propagatein the reverse direction to the record of object waves, according to Eq. (3.21). The resulting realimage of the interferogram is observed in the object plane of the lens e.g. on a ground screen.Next it can be approved that in this plane equal undistorted phase differences are created like inthe hypothetical case of the simultaneous presence of a diffuse surface in both undeformed anddeformed states. Also, it is easy to show that the impulse response of the system is independentof the orientation of the ray tracing, consequently, in focal plane of the reconstruction lens, thesystem is characterized by the expression analogous to Eq. (3.19). The reconstructed complexamplitude in the z = z0 plane is described by the superposition integral

a0(x0, y0) =∫∫ ∞

−∞a∗2(x2, y2)eik(p+q) exp

[2

ikq − f

(x22 + y2

2)]

× exp[−ikdf

(x2

q − f+x0

f

)]J1(2π%′2bf )

%′2dx2 dy2 .

(3.22)

Further, we express the amplitude a∗2(x2, y2) using the integral from Eq. (3.3) and impulse re-sponse of the system. Assuming that the impulse response of the system with a diaphragm


aperture is determined mainly by the aperture function Pf (xf , yf ), leads to P1(x1, y1) = 1 onthe whole interval. Consequently from Eq. (3.8) and Eq. (3.3) we get

a2(x2, y2) =∫∫ ∞

−∞a0(x0, y0)eik(w+p+q)eik(x2

0+y20)/2p′

eik(x22+y2

2)/2q

× 1Mδ(x2

M+ x0,

y2M

+ y0

)dx0 dy0 .

(3.23)

The two quadratic phase terms can be approximated according to Goodman [25]

eik(x20+y2

0)/2p ∼= exp[

ik2p

(x2

2 + y22

M2

)]. (3.24)

The approximation is based upon the knowledge, that distribution of the light field in the areaof the point of the image is affected only by vicinity of the optically conjugated surface area ofthe object. Eq. (3.24) is valid according to the lens equation for the geometric image of a singlepoint, i. e. the center of diffraction pattern. Then, the integral in Eq. (3.23) might be simplified byfactorization of the phase elements, and given the properties of δ-function following expressionfor the amplitude a2(x2, y2) is found

a2(x2, y2) = eik(w+p+q) exp[ ik2q

(1 +

p′

q

)(x2

2 + y22

)] 1Ma0

(−x2

M,− y2

M

). (3.25)

Substituing the complex conjugate of a2(x2, y2) into Eq. (3.22) it can be written

a0(x0, y0) =∫∫ ∞

−∞e−ikw exp

[ ik2

( 1q − f

− p′

q2− 1q

)(x2

2 + y22

)]× exp

[− i2πλdf

(xf

q − f+x0

f

)]J1(2π%′2bf )

%′2

× 1Ma∗0

(−x2

M,− y2

M

)dx2 dy2 .

(3.26)

The exponent of the second phase element in the integral is nearly zero, therefore this term couldbe neglected. We rewrite the third term to the following form

exp[−ik

df

f

(x2

M+ x0

)]≈ exp

[−ik

(x2

M+ x0

)cosβx

]. (3.27)

It expresses the same phase distribution as a planar wave propagating at the angle βx to the planeof the object surface, while its phase at the considered point x0, y0 in the object plane of the lensis zero. Substituting the complex amplitude a0(x0, y0) according to Eq. (3.2) into Eq. (3.22),and expressing planar wavefront (3.27), we get a general expression for a phase shift of a surfacepoint on the body, which was repositioned in the interval between two exposures by the vectorwith coordinates u,w.

exp [−ik(x0 cosαx + z0 cosαz)] = exp[−ik(x0 cosβx + z0 cosβz)] . (3.28)


Hence, the resulting condition for emerging of the interference fringes is

u(cosαx + cosβx) + w(cosαz + cosβz) = Nλ , (3.29)

where N is the fringe order. This expression, or alternatively for the component v analogousto u, allows us to evaluate quantitatively the interference fringe with respect to the orthogonalcomponents of the surface displacements.

As it can be seen from the expression Eq. (3.27) for the phase element, moving the apertureof the spatial filter positioned in the focal plane may be regarded as changing of the observationdirection, and it can be chosen at reconstruction. When the aperture is positioned to the opticalaxis of system, Eq. (3.29) of interference pattern interpretation is simplified

u cosαx + w(cosαz + 1) = Nλ . (3.30)

Provided that the object at the recording is illuminated by planar wave propagation along theoptical axis of the system, the next relationship will be valid for the interference fringes

w =Nλ

2. (3.31)

The resulting interferogram represents an interferogram of the out-of-plane component w sepa-rated from the in-plane components u and v.

Besides the possibility of choosing the observation direction, spatial frequencies filtering inthe reconstruction phase provides also other advantages. For example, when recording double-exposure holograms whole surface of the lens aperture is used and exposure time may not be solong. Diffuse object surface reflects only a few percent of the incident light, thus increasing ofthe effective opening diameter, seeing that the exposure time falls quadratically, can be noticable.

Other interesting features of holography include the possibility of recording and reconstruc-tion of non-distorted image of an object through optically distorting medium. When recon-structing holographic interferograms in the same optical system with imaging lens, the variousaberrations of the lens are compensated by itself. Reconstructed image on the ground screenplaced in the object plane of the lens is therefore an exact copy of the object surface with fringes.These interference fringes will be localized in the plane of the ground screen.

Nevertheless, the most important property of such double-exposure hologram is an oppor-tunity to evaluate the in-plane components of surface displacements. The separation of thesecomponents can be performed in a manner known from speckle interferometry. First of all thepointwise filtering with unexpanded laser beam can be carried out to search for Young’s fringes asit is convenient in standard double-exposure specklegram. Regardless of “parasitic”±1st diffrac-tion orders due to diffraction on the carrier frequency of holographic record the laser beam formsa central diffraction halo with fringes of well-defined fringe spacing and orientation. By point bypoint scanning of the record, complete information on the map of in-plane displacements can beobtained.

Nevertheless, whole field of in-plane displacements can be retrieved also by whole-field filter-ing method. If the recorded double-exposure hologram Eq. (3.20) is illuminated by a convergentwavefront, three diffracted beams appeare behind it. Two of them on the outer space carry infor-mation on the displacement component w and for now are off interest. The central wave – the


d

LRqR

br

pR

LK

x'2, y'2 x'R, y'R x'3, y'3

Fig. 3.2. Optical scheme of in-plane components reconstruction.

zeroth order of diffraction – is described as the second term of the brace in Eq. (3.20). Complexamplitude just behind the hologram will then be

(a221 + a2

22) exp(− ik

2pR

[(x′2 − d)2 + y′22

]), (3.32)

where the constants T0, β, t are omitted. Designation of distances and coordinates is shown inFig. 3.2 assuming the origins of coordinate systems lay on optical axis of the lens LR. Complexamplitude in the plane of lens LR from the left is

a−R(xR, yR) = exp(

ikx2

R + y2R

2pR

)∫∫ ∞

−∞(a2

21 + a222)e

−ikh(x′

2−d)2+y′2

2

i/2pR

× eik(x′22 +y′2

2 )/2pR e−ik(x′2xR+y′

2yR)/pR dx′2 dy′2 .

(3.33)

It is immediately seen that in the plane, where the reconstruction wave converges, Fourier’simage of the hologram transparent is formed. There is, therefore, an option of spatial frequenciesfiltering. Placing the lens LR with aperture PR(xR, yR) into the diffraction halo, image of theobject with interference fringes proportional to the planar components of displacement will beprojected on the ground screen.

Let the relation (3.33) be rewritten into the form

a−R(xR, yR) = eik(x2R+y2

R)/2pR

∫∫ ∞

−∞

(a221 + a2

22

)e−ikdx′

2/pR

× e−ik(x′2xR+y′

2yR)/pR dx′2 dy′2 .(3.34)

Light amplitude in the image plane of the lens LR can be found by applying diffractionintegral

a′3(x′3, y

′3) = eik(x′2

3 ,y′23 )/2qR

∫∫ ∞

−∞a+R(xR, yR) eik(x2

R+y2R)/2qR

× e−ik(x′3xR+y′

3yR)/q dxR dyR ,(3.35)


where a+R(xR, yR) is amplitude behind the lens LR

a+R(xR, yR) = a−R(xR, yR) PR(xR, yR) eik(x2

R+y2R)/2fR . (3.36)

Substituting the combination of Eq. (3.34) and Eq. (3.36) into Eq. (3.35) we get

a′3(x′3, y

′3) = eik(x′2

3 ,y′23 )/2qR

∫∫∫∫ ∞

−∞

(a221 + a2

22

)e−ikdx′

2/pRPR(xR, yR)

× exp{−ik

[xR

(x′2pR

+x′3qR

)+ y2

(y′2pR

+y′3qR

)]}× dxR dyR dx′2 dy′2 ,

(3.37)

where the lens equation for LR was used. Solving the double-integral is as follows

a′3(x′3, y

′3) = eik(x′2

3 ,y′23 )/2qR

∫∫ ∞

−∞

(a221 + a2

22

)e−ikdx′

2/pR

× J1(2π%′3bR)%′3

dx′2 dy′2 ,(3.38)

in which %′3 denotes polar radius in the following spatial frequencies

ξ′3 =1λ

(x′2pR

+x′3qR

); η′3 =

1λ

(y′2pR

+y′3qR

). (3.39)

Since registration of hologram is done by the intensity detection (photography), the quadraticphase term in front of the integral need not be taken into account. Interference fringes appear as aresult of the phase stroke in the spatial frequencies domain, which is described by the linear terminside the integral. This term can be excluded in front of integral using Goodman’s approximation(3.24). Thus at the reconstruction, the amplitude records of the first and the second exposures,respectively, are shifted in their phases so they will interfere. Intensity distribution in the imageplane x′3, y

′3 should be

I(x′3, y′3) = [a∗21(x

′3, y

′3) + a∗22(x

′3, y

′3)] [a21(x′3, y

′3) + a22(x′3, y

′3)] (3.40)

or

I(x′3, y′3) = 2a2

[eik(ϕ1−ϕ2) + e−ik(ϕ1−ϕ2)

](3.41)

while the waves are meant in a general form

a21(x′3, y′3) = aeikϕ1 ,

a22(x′3, y′3) = aeikϕ2 .

(3.42)

After rewriting Eq. (3.41) to trigonometric form we obtain

I(x′3, y′3) = 2a2[1 + cos k(ϕ1 − ϕ2)] . (3.43)


For the interference fringes

ϕ1 − ϕ2 = Nλ , (3.44)

where N is the fringe order.Applying the lens equation and Eq. (3.10) we get

x′3d

qR= −x

′2d

pR=M(1− w/p)x0d

pR. (3.45)

Then,

u+x02

pw = Nx

λpR

Md. (3.46)

Displacements in y direction can be determined in the same manner as in direction x. Theonly difference is that the aperture of the lens LR is placed near the diffraction halo edge in they axis

v +y02pw = Ny

λpR

Md; (3.47)

Eq. (3.46) and Eq. (3.47) are the basic formulas for the interpretation of interference fringesof in-plane components. The second term on the left side involves the quantity w and can becalculated on the basis of the component evaluation according to Eq. (3.31). The influence ofthis term is usually very small and it diminishes in the vicinity of the optical axis. Therefore, for asmaller object and larger diameter lens, we can achieve the interference pattern which representsfringes of constant values of in-plane u or v displacement components quite well.

An interesting and favorable feature of expressions in Eq. (3.46) and Eq. (3.47), from theexperimental point of view, is their dependence on the parameters M,d, pR. It allows to changethe interference sensitivity to a certain extent. In reconstruction the sensitivity constant

cd =λpR

Md(3.48)

may be improved/reduced by increasing the distance d. The range of measured displacements islimited from below by sensitivity of the interferometer to the smallest displacement which canbe measured (as the first order interference fringe). The best value of pR/d ratio is determinedby the dimensions of diffraction halo, when the lens aperture in reconsturction is placed near thehalo. Next we have proven that the dimension of diffraction halo is unambiguously determinedby the recording lens numerical aperure. Let us express the amplitude a′R(x′R, y

′R) in the focal

plane of the convergent wave by means of coordinates with origin on the optical axis of the lensLK (Fig. 3.2). It can be done by Fourier transformation of amplitude transmissivity functionof an image-plane hologram. Since the phase terms do not influence the intesity distribution inthe diffraction halo, they will be omitted. In the diffraction integral we substitute the amplitudetransmissivity function according to Eq. (3.9)

a′R(x′R, y′R) =

∫∫ ∞

−∞

J21 (2π%2b1)

%22

exp[− ikpR

(x2x′R + y2y

′R)]

dx2 dy2 . (3.49)


Substituting the coordinates x2, y2 by means of Eq. (3.7) (when out-of-plane component w isneglected) the integral takes the form


∫∫ ∞

−∞

J21 (2π%2b1)

%22

exp[−i2π

q

pR(ξ2x′R + η2y

′R)]

dξ2 dη2 , (3.50)

whose solution is the following function [26]


2 cos−1 r′R2b1

− r′Rb1

[1−

(r′R2b1

)2]1/2

, for r′R < 2b1 ,

0 , for r′R > 2b1 ,

(3.51)

where

r′R =q

pR

(x′2R + y′2R

)1/2. (3.52)

After the replacement of q = f(M + 1), in the illuminated area the following relationshipapplies

f(M + 1)2b1

≤ pR

(x′2R + y′2R )1/2, (3.53)

from which a clear dependence of the best ratio pR/d on the lens numerical aperture f/2b1follows. The sensitivity constant cd will be expressed according to the recent relation as

cd ≤λf

2b1

(1 +

1M

). (3.54)

It is known that well corrected lenses have a ratio of diameter to the focal length of no morethan ∼ 1/1.4. This value is the main limiting factor for the sensitivity of the interferometerlayouts. Besides using smaller wavelengths the only possibility of increasing the sensitivityis to magnify the image recorded on hologram as seen from Eq. (3.38). For example, changingthe image magnification from 1/1 to 2/1 increases the sensitivity of the interferometer 1.33 times.Theoretically, twofold increase in sensitivity with regard to the magnification 1/1 can be achieved(at infinite image magnification). Dependences of reciprocal values of the constant cd and imagemagnification on the hologram are plotted in Fig. 3.3 for different lens numerical apertures.

In all the derived relations for the interpretation of interferograms of displacement compo-nents we have assumed the expressions to be valid also for diffuse scattered coherent light. Thismeans that in the double exposure records the correlation of the recorded holograms must not beinfringed. In practice, the vicinity of points on diffusing surface are moved piecewise as a whole,because the gradients of displacement in the neighborhood are not too large, so the surface mi-crostructure is changing slightly. For the formation of interferogram is, however, neccessary thatthe considered quantities of displacements did not exceed the size of coherence area on the re-constructed image. The area of coherence in diffuse coherence radiation is equal to mean sizeof laser speckles, which is determined by the impulse response of the system, see Eq. (3.9),


0 1 2 3 4 5 6 7

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 � � � � � � � �

f � � � � b � � � � � �

f � � � � b � � � � � � �

� � c d � � � � � �

� � � � � � � � � � �

f � � � � b � � � � � � �

Fig. 3.3. Dependences of the constant cd and image magnification on the hologram [27].

Eq. (3.19) or Eq. (3.38), respectively. The relationships reflect reality more or less correctly, inpractice, however, they may sufficiently express the statistical average values.

Correlation properties of light speckles recorded during both the first and the second expo-sures, which are observed simultaneously at reconstruction, determine directly the contrast of theinterference image. Hence, the value of displacements of the surface points should not exceedthe characteristic dimensions of the laser speckles, otherwise the contrast of interference fringesis reduced to zero and the interferogram disappears. However, the interferogram could also beobtained from such records, provided that the degree of cross-correlation of the records for bothexposures was improved in the process of reconstruction. It is possible to increase the coherencearea during the component w reconstruction by reducing the aperture of the spatial filter as wellas by the appropriate choice of the diameter of diaphragm aperture of the lens LR thereby toimprove contrast of the fringes. Continuously varying aperture in reconstruction allows to selectvisually the optimal contrast and speckle structure of the image.

When observed interference patterns of in-plane displacements, the low contrast of fringesis visible at a glance even when using small aperture PR of the filtering diaphragm. The mainreason is in the lacking of any carrier frequency and the image of an object with interferogramis reconstructed via light diffraction on the hologram laser speckles in the zero order of diffrac-tion. The quality of these fringes could be substantially improved using double-aperture (oralternatively four-aperture) diaphragm of the imaging lens. This will create a double-aperture in-terferometer type, which provides more contrast interferograms of in-plane components. By suchmeans the double-aperture type interferometer is built, which provides more readable interfero-gram of in-plane components [28, 27]. The basic principle of creating an interference pattern inthis interferometer is very close to the standard speckle scheme with spatial frequencies filtering,as described by relations Eq. (3.38), Eq. (3.46), Eq. (3.47).

Optical layout of the double-aperture type interferometer is shown in Fig. 3.4. The aperturediaphragm C with two or four circular openings is placed in front of the lens L with focal lengthf . Using four aperture openings allows simultaneous recording of both in-plane components u


d'

LC

x0, y0

b'x1, y1 x2, y2

O'

P

z

p q

2d'

2b'

Fig. 3.4. Double-aperture type interferometer for in-plane components of displacemets measurement.

and v. The distance of two holes is 2d′, their diameter is 2b′. Coherent wave that illuminatesthe object is either reflected from the opaque diffusing surface, or passes through the transparentmodel M0 and subsequently is diffused by its groud-screen-like face. The transparent schemewith one diffused object was used to record the isothetics. Isothetics are the fringes of the samevalues of in-plane displacements. The basic idea of the interferometer comes from the wavefrontdivision emanating from the points on diffusing surface into two parts. If the lens aperture isscreened only by one opening, the image of diffuse surface on the holographic plate created incoherent light consists of a variety of irregular interference maxima and minima – laser speckles.When light passes through both the openings of a double-aperture diaphragm, the wavefrontsfrom the first and the second apertures are impinging on a holographic plate at certain angle, sothey interfere. The resulting image of laser speckles is modulated by periodic grating. Illuminat-ing such record, image of the diffuse surface can be reconstructed.

Based on diffraction theory we derive the creation of holographic recording and the formationof the interferogram in displacement fringes measurement. Impulse response of the system isexpressed in Eq. (3.8) with the following conditions: the double-aperture function equals tounity in circles of radius b′ distanced from the optical axis by ±d′. Complex amplitude of thediffraction pattern in the plane z = z0 is

a′2(ξ2, η2) =J1(2π%2b

′)%2

ei2πd′ξ2 , (3.55)

where the constant terms were omitted. The same procedure gives complex amplitude for thesecond aperture opening and the resulting amplitude of the both circles will be the sum of both

a2(ξ2, η2) =J1(2π%2b

′)%2

(ei2πd′ξ2 + e−i2πd′ξ2

). (3.56)

Intensity distribution of the diffraction pattern is the squared amplitude

I(%2) =J2

1 (2π%2b′)

%22

(2 + ei4πd′ξ2 + e−i4πd′ξ2

)(3.57)


and finally, in trigonometric form

I(%2) =J2

1 (2π%2b′)

%22

cos2(2πd′ξ2) . (3.58)

This result can be interpreted as diffraction pattern of one of the circular openings modulated bycos2(2πd′ξ2). Thus, in the diffraction pattern the distribution of intensity is formed, which cor-responds to the linear so-called sinusoidal grating with the fringes perpendicular to the flowlinebetween the aperture holes. The grating frequency f0 is determined according to Eq. (3.58) bytwo consecutive zero values

f0 =2d′

λq. (3.59)

Double-exposure record on photographic material with linear characteristics can be consid-ered as the transparent with amplitude transmissivity

T ′2 (x2, y2) = T ′

21(x2, y2) + T ′22(x2, y2) , (3.60)

where T ′21(x2, y2)+T ′

22(x2, y2) are the transmissivities of the first and the second exposures. Theproper reconstruction scheme of such record is analogous to the convential filtration in Fouriertransform plane (Fig. 3.2). The diffraction field in the hologram plane is

a′2R(x2, y2) = e−iπ(x22+y2

2)/λpR T ′2 (x2, y2) . (3.61)

Further we substitute transmissivities T ′21(x2, y2), T ′

22(x2, y2) by means of Eq. (3.57), where weassumed the coordinates in the first and second exposition to be (x01, y01, 0) and (x02, y02,−w).Let us consider wavefronts observed in +1st or −1st order of diffraction

a′2R(x2, y2) =J2

1 (2π%21b′)

%221

ei4πd′(x2+Mx01)/λq

+J2

1 (2π%22b′)

%222

ei4πd′[x2+Mx02(1−w/p)]/λq .

(3.62)

The phase term has to be taken into account.As the component of displacement u = x02 − x01 is small compared to x01, x02 after some

modification of Eq. (3.62) we get

a′2R(x2, y2) =J2

1 (2π%2b′)

%22

ei4πd′(x2+Mx01)/λq[1 + e−i4πd′(u+x02w/p)/λq

]. (3.63)

In trigonometric form

a′2R(x2, y2) =J2

1 (2π%2b′)

%22

cos[4πd′

λq(x2 +Mx01)

]×2 cos2

[2πd′

λp

(u+

x02

pw

)].

(3.64)


(a) (b)

Fig. 3.5. Interference fringes of equal values of in-plane diplacement components around the central crackin loaded three-point-band beam. (a) hroizontal or x component u of displacement vector, (b) vertical or ycomponent v of displacement vector [29].

Fig. 3.6. Isothetics of horizontal in-plane component of displacement vector. PMMA model of wall wasvertically loaded by constantly distributed load.

In this relation, as seen, the second factor represents sinusoidal carrier frequency and the lastfactor describes the interference fringes of constant displacements. Considering conditions formaxima as well as minima we obtain

u+x02

pw = N

λp

2d′. (3.65)

The displacements in the direction of y axis are determined similarly by orienting the con-necting line of the holes parallel to the axis y. Interference fringes of contstant components ofdisplacements u, v are presented in Fig. 3.5 and Fig. 3.6.

A more detailed description of the double-aperture type interferometer, its characteristics,advantages and drawbacks are in the work [27].

3.1.1 Experimental application of interferometer

The common equations for the interpretation of fringes were verified experimentally and usedto evaluate the measurement of fringes of displacement components in the beam model withhalf-elliptical crack. Besides the confirmation of derived theoretical relations, the experimenthad to show in particular the practical applicability of a similar interferometer in ExperimentalMechanics.


Optical interferometer setup was built using conventional optical elements: He-Ne laser,collimator lens, imaging lens, beam-splitter mirror and planar mirrors. The laser beam wascollimated, part of which passed through the beam-splitter mirror and another part was reflected.The part of beam passing the beam-splitter was further directed to the holographic plate andserved as a reference beam. Diffuse object surface was illuminated by collimated beam and wasdisplayed via imaging lens on the holographic plate. A He-Ne laser with relatively low power10 mW and photo-plate Agfa E75 were used in experiments. The imaging lens had focal lengthof 200 mm.

One of the basic experimental problems, occuring almost always in double-exposure opticalmethods, is to ensure immobility of the sample as a whole when straining in the interval betweenexposures. Uncontrolled movement of the sample as a whole affects the resulting interferogramsobtained, which can therefore be considerably distorted. When recording complete informationon the deformation and movement of the body, as it is the case with simultaneous record ofall three components of the displacement vector, the issue is of secondary importance, sincethe distortion only slightly complicates the data processing and does not affect the evaluation ofelement deformation. Incomplete knowledge of information on displacement vector (e.g. specklemethods) can cause significant inaccuracies in the evaluation, as pointed out e.g. in [1, 30].

It is always preferable, also in terms of simplifying the data processing, to make sure that thechosen coordinate system on the object remaines stationary during loading. Frequently occurringpractical problems are symmetrical samples where the origin of coordinates is to be chosen onthe plane of symmetry. To ensure a better carrying out such experiments, we also developeda load frame, in which, in contrast to conventional constructions, both load points are moving.Thus, the center of the symmetrical body stays immobile during deformation. Working principleof the load frame is based on hydraulic cylinders with pistons providing tensile forces on bothsides. Load forces are measured by the manometer showing the pressure in the cylinders. Axisof load pistons moves in special friction bearings allowing very accurate guidance with accuracyin order of magnitude of light wavelength.

One of the first applications where we used the interferometer for simultaneous recordingof all the components of displacement vector together with the described loading frame was themeasurements of displacement vector on the beam model with semi-elliptic notch imitating anarrow crack. An example of experimentally obtained interference patterns of isothetic fringesis shown in Fig. 3.7. Images on the left show the front wall of the beam with a crack, rear wall ison the right. The top two interferograms represent lines of equal values of the component w per-pendicular to the displayed wall; the middle two are the interference fringes of the component uparallel to the longitudinal axis of the beam and at the bottom there are figures for the componentv perpendicular to this direction.

In conclusion both advantages and disadvantages of the developed holographic-interferenceprinciple of simultaneous recording and optical separation of displacement components are brieflysummarized. The proposed method makes possible (in the basic variant):

• Interferogram record of all three components of the displacement vector to one hologramin one loading process

• Direct optical recording of the components’ interferograms in the reconstruction

• Simplification of the experimental procedure as well as, due to the information recording


Fig. 3.7. Fringe patterns of equi-lines of out-of-plane displacement component w, and orthogonal in-planecomponents u, v.

in a single load process, considerable reduction of potential measurement errors comparedwith separate entry for each component, or records and reconstruction under the classicalAlexandrov and Bonc-Brujevic scheme

• Visual selection of the fringes contrast in the reconstruction compromising between thefringes contrast and acceptable graininess of the image

Disadvantages of the method include

• Restriction to relatively small objects comparable to the size of the imaging lens

• Lower sensitivity in the planar components which is umin = vmin ≈ 2λ in comparisonwith the sensitivity of the component wmin ≈ λ/2

Methods of experimental stress analysis allow us to determine stress components on the sur-face as well as in the interior of solids by evaluation of surface displacements. It is, however, notpossible always to evaluate both the displacement and stress fields by experimental measurementof the object deformation only. As a rule, it becomes necessary to use analytical expressions ofconstitutive relation or to utilize some kind of numerical methods to supplement experimentalresults. The combination of finite element method (FEM) or boundary element method (BEM)with an experimental measurement of deformations is a promissing solution of the problem. TheBEM for the stress analysis is the method suited very well to solve two- and three-dimensional


5r =16

40

10

σ

σ

Fig. 3.8. Configuration of the beam specimen with semielliptical part-through surface crack.

problems as it reduces them to boundary solutions, i.e., only elements on the boundaries needto be defined. Depending on the completeness of the experimental data, we first studied the un-known reactions and/or displacements on the boundary of the studied region or we proceededimmediately to the calculation of the stress field at internal points of the region [31, 29].

The hybrid numerical and experimental method for the analysis of three-dimensional prob-lems permits the determination of stress components at interior points non-destructively. Themethod has another advantage comparing purely numerical approach by allowing a reduction ofthe initial complicated problem to a problem of simple boundary/surface conditions. Moreover,it takes into account the real boundary conditions, i.e. with the friction between mechanicallyinteracted bodies, temperature loading or other ambient conditions.

To demonstrate the hybrid experimental and numerical approach, we have chosen a three-point loaded beam with an edge crack and also the model of a large slab wall stiffened by aframe. The displacement fields of the region studied are displayed in Fig. 3.5 and Fig. 3.6.Both of these cases represent two-dimensional problem of plane stress. As a next example, suit-able to demonstrate the three-dimensional stress state problem by the proposed combination ofexperimental-numerical method, a beam subjected to uniformly distributed tension was chosen,including part-through crack (see Fig. 3.8). The model made of PMMA was loaded and observedfrom four faces by holo/speckle interferometer. Recorded three orthogonal displacement vectorcomponents are in Fig. 3.7.

The experience gained with applications of the hybrid experimental-numerical method inmechanical stress state problems solution over the years has demonstrated the effectiveness ofthis approach and simplified the use of labor-consuming evaluation of experimental data [32].

3.1.2 Two-channel speckle interferometer

In view of its simplicity, the speckle method is very convenient, but during its exploitation in amechanical laboratory the the insufficient sensitivity to in-plane displacement detection has beenascertained as a strong limiting factor. Even in the electronic speckle shearing interfereometrymathematical fitting must be used to interpolate the fringe positions data. In order to improvemeasuring sensitivity as well as some other qualities, we have proposed [33] new simple opticalarrangement of the speckle camera. Unlike the single-aperture photography setup, the double-exposure speckle/holograms are recorded by a two-channel optical scheme (Fig. 3.9). Two im-ages of the observed surface are projected independently onto the photographic plate throughthe reflection on the two side mirrors. By appropriate focusing, both the wavefront are imping-


HologramObjectivelenses

MirrorsObject

Fig. 3.9. Optical scheme of double-aperture two-channel holo/speckle interferometer [33].

(a) (b)

Fig. 3.10. In-plane displacement components (a) u and (b) v of a nuclear reactor vessel wall with crack.

ing on the photoplate at mutually opposite angles and create carrier frequency of the speckles,consequently, high contrast interference fringes are generated without need of additional opticalfiltration.

In the optical setup, the viewing angle can be enlarged 3 to 5 times comparing to standardspeckle interferometry scheme with wide-open lens. As the carrier frequency of the speckles issufficiently higher compared to the double-aperture speckle scheme, two-channel scheme pro-vides greater opportunities how to depress decorrelation effect but also how to realize the phaseshift in electronic speckle interferometry variant. Other advantages lie in smaller geometricaldistortion of the image, and also in better conditions for measuring of large structures. The de-veloped interferometer was used e.g. in experimental testing of the thermally loaded stressesacross the layered structure of nuclear reactor vessel wall with crack. Fig. 3.10 shows the mea-sured displacement field of both the orthogonal in-plane components.

3.2 Reconstruction of records in speckle interferometry using polychromatic light

Important place among the methods of so-called speckle interferometry belongs to the double-aperture-type interferometer. Its advantage lies mainly in the simplicity of experimental equip-ment and direct recording of fringes of equal planar components of displacement vector. More-over the quality of interferograms is much better than the fringes obtained by Fourier filtrationof classical specklegrams. It is a consequence of the fact that records in double-aperture-type in-terferometer feature carrier frequency in contrast to the usual specklegram. In fact, this blurs the


difference between double-aperture interferometer and imaging hologram records respectively,and specklegram-hologram has many properties of imaging hologram. In terms of interferome-try, the most important is the possibility of white-light reconstruction as well as precise fringeslocalization on the surface of the measured object.

The essential element of double-aperture-type interferometer is a lens with double-aperturebarrier in front of it. Diffraction pattern of the two waves from the apertures is recorded onphoto-plate in the image plane of the object. Diffraction pattern of a single point coherent sourceis given by Fraunhofer diffraction of the appropriate aperture shape function [26, 27]. Then theintensity distribution can be written in the form

I(%) = [J1(2π%b)/%]2 [2 + exp(i4πdξ) + exp(−i4πdξ)] , (3.66)

where

ξ =x2 +Mx0

λ0q0; η =

y2 +My0λ0q0

; % =√ξ2 + η2 , (3.67)

λ0 is wavelength of light on the recording plane and M = q0/p0 represents object magnificationon the photo-plate.

In the reconstruction scheme of such linearly recorded hologram-specklegram, Fourier’sspectrum of the transparent hologram-specklegram appears in the camera lens plane (x3, y3)The transmittance of the transparent is

T2(x2, y2) = T21(x2, y2) + T22(x2, y2) , (3.68)

where

T21,22 = KtI21,22(%) (3.69)

are the records obtained during the first and second expositions, respectively, in the experimentof fringes of equal in-plane displacements observation. The symbol K denotes proportionalityconstant of the photomaterial and t is the exposure time.

The amplitude of the light field in (x3, y3) plane, having neglected the constant coefficientsthat do not affect its spatial distribution, can be expressed as follows

a3(x3, y3) =∫∫ ∞

−∞T2(x2, y2) exp

[−i

2πλq

(x2x3 + y2y3)]

dx2 dy2 . (3.70)

Substituing T2(x2, y2) according to Eq. (3.66), (3.68) and changing coordinates x2, y2 by meansof Eq. (3.67), the amplitude takes new form

a3(x3, y3) =∫∫ ∞

−∞(J1/%)2 exp(−i4πdξ) exp [−i2π(fxξ + fyη)] dξ dη , (3.71)

where

fx =λ0q0λq

x3; fy =λ0q0λq

y3 , (3.72)


while just the second term from the bracket of Eq. (3.66) was taken, which represents the +1st

diffraction order. Let the Fourier transform of the function (J1/%)2 to be denoted as F(J1/%)2.Then, according to the shift theorem, a3(x3, y3) represents the same function shifted in the sys-tem of coordinates by

x30 =λq

λ0q02d . (3.73)

The last equation shows that the change in wavelength of light λ in the plane distanced fromthe transparent by q expands the spectrum, the position of the diffraction halo varies with thewavelength of reconstructing wave. Consequently, when using reconstructing white-light source,in the ±1st orders the diffraction halo is decomposed into light spectrum. Nevertheless, as it willbe shown below, the actual interference pattern that is created by following transformation in the(x4, y4) plane is not affected by this decomposition.

For simplified notation, assume that the center of the camera lens is located on the opticalaxis. Distribution of the light field after passing through the lens is

a′3(x3, y3) = a3(x3, y3) exp[− iπλf

(x23 + y2

3)]. (3.74)

Amplitude of wavefront incident on the film in the camera a4(x4, y4) can be obtained usingFresnels’ diffraction equation

a4(x4, y4) =∫∫ ∞

−∞a′3(x3, y3) exp

[iπλq′

(x23 + y2

3)]

× exp[− i2πλq′

(x4x3 + y4y3)]

dx3 dy3 .(3.75)

After substituting from Eq. (3.74) and applying simple operations we get

a4(x4, y4) =∫∫ ∞

−∞a3(x3, y3) exp

[− iπλq

(x23 + y2

3)]

× exp[− i2πλq′

(x4x3 + y4y3)]

dx3 dy3 .(3.76)

While x23 + y2

3 � q, the term exp[− iπ

λq (x23 + y2

3)]

can be set as 1 in the first approximation.

Whereas, according to Eq. (3.73), function a3(x3, y3) represents function F(J1/%)2 shifted inthe coordinate system, the transformation Eq. (3.76) can be treated again using shift theorem.Then,

a4(x4, y4) = (J1/%)2 exp(i4πdξ) . (3.77)

It is seen that the amplitude distribution in the (x4, y4) plane is independent of F(J1/%)2 shift inthe (x3, y3) plane.

For double exposure process of moving point-like source in the object plane we get the re-sulting interference pattern by summing the amplitudes a4(x4, y4). In this case, it is necessarythat the displacement does not exceed the radius of correlation, which is roughly the radius % ofthe first diffraction minimum in the envelope surface of the modulation grid peaks described bythe Eq. (3.66).


(a) (b)

Fig. 3.11. Interference fringes of equal values of in-plane diplacement components (a) u and (b) v of thewall with window loaded in vertical direction by constantly distributed load.

3.2.1 Application conclusions

Fig. 3.11 shows the interferogram of the displacement components on the model of wall. Fromthe reconstructed image in white light it is easy to see that the characteristic speckle interfero-gram image noise is strongly suppressed. This improves resolving power and accuracy of theindividual fringes identification.

Moreover, in order to ensure non-distorted interferogram reconstruction in the case of coher-ent light a precise location of the camera lens aperture sensing the middle of the ±1st diffractionorder halo is required. Using white-light reconstruction eliminates this problem and the interfer-ence fringes are localized on the object surface – its shape does not change with the change ofthe position of the objective lens. What is more, it is also possible to use non-point source oflight (eg. projector). It should be remembered, however, that polychromatic light reconstructsonly the records with correlated exposures. This implies the use of double-aperture element withrelatively small apertures while recording, which is certainly a disadvantage with regard to theneed for longer exposure time.

3.3 Speckle interferometry by sandwich principle

Classical double-exposure speckle interferometry is known as relatively simple and effectiveoptical method. It is, however, often influenced by adverse factors. As the first is the fact that indouble-exposure on one photo-plate the overall displacements of the observed surface in spaceare recorded. In technical applications these displacements often exceed the deformation ofthe object resulting in significant inaccuracies in evalution. Another limitation of the classicspeckle scheme is narrow range of measurable values of displacements and in particular limitedability to record the smallest values. This is, at best, an order of magnitude worse than in theholographic interferometry measurements of out-of-plane displacement. Another fact must betaken into account that in classical speckle interferometric scheme a record objective lens withlarge aperture is necessary to increase sensitivity of the measurement. Most lenses with suchapertures are characterized by large geometric aberrations leading to significant measurementerrors [34].

Using so-called sandwich principle [35] can solve problems mentioned above. The method isbased on the phenomenon of interference of light diffracted on two identical structures arrangedone behind another. Each of the exposures recording object in two states of deformation is done


Fig. 3.12. Interference pattern of two specklegrams positioned one behind another.

h

ℓ1

ℓ2

12

ℓ'2

ℓ'1

n

βrx

Fig. 3.13. Interference scheme of two point-like light sources placed one behind another.

on a separate photo-plate. In the information evaluation, the processed photo-plates are put ontoeach other with emulsions in one direction. When illuminating such a composition, correspond-ing pair of speckles represent a pair of consecutive – in the distance the thickness of photo-plate –point-like light sources. Diffraction on such speckle structure is a circular diffraction halo mod-ulated by fringes in the form of concentric circles (Fig. 3.12). Figure 3.13 shows the interferencescheme of two point-like light sources placed one behind another. Intensity distribution can befound when the difference of the optical pathes of beams 1 and 2 is expressed

∆ = n(l1 − l2) , (3.78)

assuming that

l′1 = nl′2 , (3.79)

where n is index of refraction of the photo-plate glass; the paths labels are in Fig. 3.13. Now weexpress l2 by means of Snell’s law for beam 2

l2 =rx sinβn

+h

ntanβ

(n

sinβ− sinβ

n

), (3.80)

while

l1 =√r2x + h2 . (3.81)


Symbol rx denotes distance of the second point from the optical axis, h is photo-plate thicknessand β is the observation angle of the considered point in the interference pattern. Since thedistance of the ground-screed is large compared to rx, the same angle β is considered for boththe rays: cosβ = 1, rx � h. Light intensity on a ground-screen point is given by interferenceof beams 1 and 2 and can be expressed using Fresnels’s formula. Provided that the amplitudesof the beams are equal, for the interference pattern applies:

I = I0 cos22πλ

(h

nsin2 β − rx sinβ

), (3.82)

where I0 is a constant and λ is wavelength of light. Hence follows the relation for interferenceminima or maxima, respectively, from which the quantity of mutual in-plane displacement of thepoint-like sources is expressed

rx =h

nsinβ − Nλ

sinβ, (3.83)

where N is interference order.In order to increase the accuracy of measurement, several points of interference pattern cor-

responding to one object point for different N and β ought to be taken into account. Sufficientlyprecise result is obtained by averaging of values rx given by Eq. (3.83). The disadvantage of theprocedure is its elaborateness – large amount of data have to be read out from the interferencepattern. Therefore, simplified method for displacements evaluation is measurement of shift ofconcentric circles centers. The position of the center is

rx =h

nsinβ , (3.84)

where N = 0 is was in Eq. (3.83). It is convenient to relate this shift to the reference positionof the center determined by the object point measurement with zero displacements. Technically,we provided the measurement using diaphragm with large circular opening and ground screenwith millimeter 2D scale. The centers of circles shift were determined relatively to the reticleset for the point with zero components of displacement. The accuracy of such measurementis relatively high due to the opaque transition of the circle on the ground screen. The uncer-tainty of circle center position was estimated to ±1 mm, which for the screen – specklegramdistance (1500 mm), photo-plate thickness 1.5 mm and the image magnification 1/3 leads to thedisplacement of object points uncertinty to be ±3 µm.

It is obvious, that in the described method the measured values of displacements are notbounded from below, even smaller quantities than the avarage value of diameters of laser specklescan be obtained. This is not possible in the classical scheme of Young’s fringes evaluation.Sensitivity of method utilizing sandwich principle is determined only by the possible error ofmeasurement of the fringes geometric position.

Moreover, in the sandwich method, numerical aperture of the objective lens has no effect onthe limitted sensitivity of measurement. Therfore, the lens can be screened to create significantlymore precise image.


3.4 Speckle interferometry utilizing digitization of image

Holographic and speckle interferometries allow contactless deformation measuring of micro-scopic objects with diffusely reflecting surface. This unique capability can be used in manyareas, however, it has limits and drawbacks. Considerable sensitivity to disturbances of ambient,such as light conditions and mechanical vibrations as well as complicated evaluation of the in-formation received are the most significant. These factors limit today’s holographic and speckleinterferometry methods are practically limited to the optical conditions of the laboratory.

Some of these drawbacks can be avoided using a photoelectric light signal sensing – thiseffort, however, encounters various problems:

• photographic plate records simultaneously the whole field of view; using one discrete pho-todetector, one loses this advantage of wide field interferometry

• the use of CCD matrix or classic vidicon would solve the problem radically, but achiev-able resolution of the pixel elements of the matrix remains at least an order of magnitudelower than the value required by the microstructure of the interference pattern forming ahologram.

3.4.1 Electronic speckle interferometry

Addressing the key problem of creating double-exposure interferogram at low resolution of therecording medium was actually already found in 1971. Butters and Leendertz [36] used pho-todetectors to measure changes in the light intensities of individual spots of the speckle structure.The main principle consists in the fact that the effective areas of the discrete sensors – CCD arrayelements was comparable to the size of the each spot. Typical statistically averaged dimension ofspots, when imaging by lenses, have a size of 5 to 100 µm, and can be controlled by appropriatenumerical aperture of the imaging lens. These dimensions are distinguishable by classic vidiconTV camera and with camera based on CCD elements.

Since the holographic interferometry is a comparison of two object states that do not existat the same time, an electronic recording storage device should be used. Given that there isalways speckle structure recorded, the method is often referred to as Electronic Speckle PatternInterferometry (ESPI).

The problem of the detection of light intensity variations of the individual spots is that it can-not be universally used in a variety of different optical schemes in holographic interferometry.A wide range of different configurations of optical assemblies has arisen as a result of the ef-fort to simplify the evaluation of information on the orthogonal components of the displacementvector on the surface of the object under examination [37]. Besides the classical scheme of Fres-nel holograms, often disadvantagous in terms of accuracy and complexity of the evaluation, theknown schemes more or less differed from one another by appropriate choice of illumination an-gles and angles of observations and recorded only displacements in the direction of observation(out-of-plane), or in a direction perpendicular to the direction of observation (in-plane). Mea-surement of in-plane displacements without use of the reference beam is usually called speckleinterferometry. In practice, generally the smallest problems of interferometry are in out-of-planedisplacement measurements, in this aspect the holographic interferometry is closest to the classi-cal interferometry, which enables such measurements only. The sensitivity of the measurements


Referenceplane

Beamsplitter

Objectivelens

Object

Fig. 3.14. Michelson interferometer, the reference wave is reflected from a reference diffusion surface,therefore its wavefront is very complex.

Object

Objectivelens

Beamsplitter

Spatial filter

Fig. 3.15. Hologram with spherical reference wavefront.

is λ/2 (the discrete step between two successive interference fringes).When using the photoelectric signal sensing there are two basic principles of interferometers

recording out-of-plane displacements. Both are based on the principle of a Michelson interfer-ometer, where two mutually interfering light wavefronts are propagating along one axis. In theformer case the information wavefront from the object interferes with the one reflected from areference diffusion surface forming a very complex wavefront (Fig 3.14). In the latter case, thereference beam is a smooth spherical wavefront (Fig 3.15).

In the former case, the contrast of obtained interference fringes is very low and the range ofthe measured values of the normal (to the surface) displacement is quite limited by decorrelationof speckle structures in larger displacements. However, this method can be successfully applied,provided that the in-plane displacements of the object deformation are relatively small comparedto the out-of-plane displacements component. Then, the mutual decorrelation of the comparedrecords is minimal and resulting contrast of interference fringes is usually sufficient. A typicalexample of such a situation in mechanics is the bending deformation of thin plates. By using themethod of electronic speckle correlation interferometry a number of mode shapes were observedunder the dynamic excitation of the plate vibrations. A powerful loudspeaker was driven by


PC

LaserMirror

Mirror

ObjectDifuser

BeamsplitterCCD

Fig. 3.16. Schematic drawing of the Electronic Speckle Correlation Interferometry intended to observe themechanical vibrations of bending plate.

tuned harmonic generator in order to excite non-contact bending vibration of the metallic plate.The idea of so-called time-average method was established shortly after the discovery of

holographic interferometry [38]. If the object during the period of exposure vibrates with periodof much less than exposure time, the multiple object states are registered on the holo/specklegram.Varying object deformations are recorded as a change in the intensity. The contribution to thetotal exposure of individual states of the object depends on the speed at which the object passesthrough these positions. On the time-average hologram interference maxima will match the im-mobile nodal positions points. The expression for the light intensity recorded in one frame is [39]

I1 = I0 + Ir + 2√I0IrJ0(P ) cosϕ , (3.85)

where P = (4π/λ)w0, w0 is the maximum amplitude of the sinusoidal vibration, J0(P ) isBessel’s function of zero order.

To enhance the contrast of the resulting interference fringes, the first two terms in Eq. (3.85)should be eliminated. For this purpose, the speckle field of the object at static nonvibrating statewas firstly recorded and then record of vibrating object surface at its resonant frequency wascarried out. Subtracting of these two frames by image processing procedure, the fringes of nodesand antinodes are visualized with irradiance proportional to [1− J0(P )]2 [40].

The test object – bending plate – was illuminated by an expanded laser beam of red lightemitted by 50 mW output power HeNe CW laser (Fig. 3.16). A part of wavefront of the extendedilluminated beam was separated by using a mirror and directed through another mirror to a smallgroundscreen. Passing through the groundscreen, the light creates the speckle field directed bythe semitransparent mirror into the objective lens of the CCD camera. The light reflected from theobject surface is passing through this beam splitter and is projected onto the CCD matrix, wherethe interference of both the speckle fields happens. Any movements of object surface along theline of sight will create the changes in optical paths and thus will give rise to the pattern ofinterferogram. The fringes of the pattern are acquired by time-average exposure and processed


(a) (b)

Fig. 3.17. (a) Vibrating mode visualization of the thin metallic plate at resonant frequency 247 Hz, (b) nu-merical simulation of the same modal shape computed by COSMOS.

Fig. 3.18. The interference pattern of the tyre sidewall bulging shape vibrating at 3rd order radial mode ofvibration – 141 Hz. Rubber tyre MATADOR Model MP 15.

in a PC by electronic substitution of both the original image speckle field and the image specklefield after deformation of object surface, caused by vibration.

The visualisation of modal structure is clear and gives us the imagination about differentmodal shapes. As an illustrative example, one of the mode shapes is shown in Fig. 3.17(a).At the right hand side in Fig. 3.17(b) there is also numerical simulation of the plate vibrationobtained by COSMOS SW tool. The dark points in the interferogram represent nodes, brightpoints represent areas where amplitude of vibrations reaches approximately the value of λ, whenλ = 632.8 nm is a wave length of the laser used and N is an integer.

Another example, where ESPI was used, is the visualization of modal shapes structure ofcar tyres. From the viewing angle of mechanics the pneumatic tyre today is a highly sophis-ticated engineering structure, where viscoelastic, anisotropic and nonhomogeneous propertiesmake reliable theoretical analysis extremely difficult. Neither the functional nor performance re-quirements can be adequately satisfied without a short sufficient understanding and knowledge ofthe strain and stress states developed within the entire composite structure under varying serviceconditions. In order to render such a study possible, an experimental approach was implemented.

Using the procedure of time-averaging the distribution of both vibrating surface amplitudesand corresponding knots was visualised (see Fig. 3.18). The entire spectrum of modal frequen-


α

A

x

x'

(1)

(2)

0θ A'

Fig. 3.19. Optical scheme of holographic interferometer providing measurements of in-plane displace-ments. The speckle field of two symmetric illuminating waves is completely non-sensitive to the objectdisplacements in the optical axis.

cies of both longitudinal and radial modal shapes up to 6.order was obtained by optical elec-tronic speckle correlation technique [41]. The vibration was excited acoustically by powerfulloudspeaker at the same ESPI setup as in Fig. 3.16. The positions of nodes and anti-nodes wereconfronted with that identified by contact mapping of the surface by means of piezoelectric ac-celerometers.

In the methods where the reference beam wavefront is in the form of complicated specklefield the fringe quality is limited by a size of individual speckle and mainly by lateral movementsof the object surface between exposures. The in-plane displacements are responsible to the effectof mutual decorrelation between both of the speckle records and consequently to the deteriorationof interference fringes contrast. In this respect it is more advantageous to use the other schemewith a smooth reference wavefront. The optical scheme has to be adjusted in such a way that thevirtual image of the point source of the reference wavefront will be matched with the center ofthe imaging lens aperture through the semitransparent mirror. The obtained holograms as a rulehave better contrast (see Fig. 3.15). The technical problem is the increase of lateral resolution ofthe image (not sensitivity), i.e. finer speckle structure and also providing more distinguishableinterference fringes (e.g. the average time method for the measurement of mechanical vibra-tions). Moreover, from a practical point of view, the mentioned optical assembly is fragile, bulkyand sensitive to vibrations.

As seen, it is possible to record out-of-plane displacements using a camera equipment. In ex-perimental practice – especially in the field of experimental mechanics – there is often a demandto monitor the displacements in the plane of surface perpendicularly to the line of sight. Unlikeclassical interferometry, holographic interferometry allows such measurements. Scheme of theinterferometer for in-plane displacements [42] is shown in Fig. 3.19. The object is illuminatedby two collimated light beams in a symmetric scheme. Each of the beams forms microscopic


θ

Objectsurface

θLaser illumination

Mirror Objectivelens

Diaphragm

CCD

Fig. 3.20. Optical scheme of laser speckle correlation interferometer.

speckle structure in the image plane where the detector is placed. It is easy to imagine that thephase changes, when the object moves along the optical axis, are of opposite sign for both ofspeckle structures, thus are cancelling each other. Only the diffuse object displacements perpen-dicular to the optical axis are interferometrically recorded. Such an interferometer can be usedin combination with CCD matrix detection. However, since in this case two mutually specklestructures interfere, the contrast of the the obtained interference fringes is very low and the rangeof measured displacement is also narrow.

The optical scheme of speckle interferometry for in-plane displacement measurement as itis drawn in Fig. 3.19, can be adapted to ensure more simple technical design. We tried to ap-ply an optical method of such electronic speckle correlation interferometry (ESCI) as a tool forvisualization of object deformation induced by moisture and temperature changes in porous ma-terial. In view of possibilities to map the space and time distribution of unfolded deformation, itis believed the method can be used to study the phenomena of mass and heat transport.

An optoelectronic system of CCD camera with digitizer and PC is used to perform digitalimage processing and appropriate correlation. The scheme of an interferometer is in Fig. 3.20.Through the reflectance on the side mirror, perpendicularly to the object surface, its diffuseplane is illuminated by two collimated light beams in a symmetric manner. Both the beamscreate independent speckle patterns that interfere after passing through an objective aperture andgenerate a resulting grained intensity variations on the CCD matrix area. From the spacing ofcorrelation fringes in-plane strain can be evaluated. The fringe value of the measurement is

cuv =λ

2 cos θ, (3.86)

where λ is the wavelenght of light.In the laboratory setup we used 650 nm/12 mW coherent laser diode. The objective lens with

focal length of 25 mm was diaphragmed 1/5.6. At such an aperture CCD chip Sony ICX039 with1280× 1024 matrix can resolve the individual speckles.

It can be said, the ESCI in such a configuration is robust sufficiently and simply applica-ble in laboratory conditions. The roughly porous surface of building materials invokes specificlimitation with the decorrelation effect between both the exposures. The effect of change in sur-face microstructure arose from structure changes caused by water filling of surface pores. Thephenomenon of the water appearance in the surface is followed by speckle pattern decorrelationand clearly marked on the surface with such a disrupted microstructute nearly water line (see


Watersurface

Mirror planeMirrorplane

Watersurface

Fig. 3.21. In-plane displacement components u (left), v (right) of the banding beam deformation inducedby water suction on the bottom beam surface, the fringe value was 0.73 µm.

d

LD

x0, y0

bx, y

x1, y1

G

p q

2d

2b

Fig. 3.22. The scheme of double-aperture speckle interferometer. Lens L is covered by double-aperturediaphragm D. Linear grating G is placed in the image plane of L with grids oriented perpindicularly to thejoining line of the aperture cicular openings.

Fig. 3.21). Thus, the points where the moisture is penetrated happen to be practically unob-servable. The effect is observable in Fig. 3.21(a) where the suction of water was measured onfreestanding beam made of porous material of AAC concrete. The contrast of interference fringescontinually deteriorates upwards from the contact of the beam with water level. If initially drymaterial is brought in contact with liquid water, water is sucting into the material. Subject to gra-dients of relative humidity, an element of wetted porous medium gaines fluid mass and deformsdue to internal capillary pressure. In Fig. 3.21 there are orthogonal displacement componentsof the bending beam deformation by capillary pressure. Interesting effect on the water level isvisible as a mirroring of the interference pattern.

3.4.2 Double-aperture speckle interferometer with electronic record

Optical scheme of the interferometer is drawn in Fig. 3.22. The lens L is covered by the di-aphragm D with two circular apertures. Their diameter is 2b, the distance of their centers is 2d.Coherent wavefront illuminating object O is diffusely scattered on its surface. The grating G is


placed in the imaging plane of the lens L. Grid lines are oriented in a direction perpendicular tothe line joining the centers of the aperture openings D. Interference pattern is sensed by a CCDcamera focused on the grid G.

The basic principle of the interferometer is based on a division of wavefronts originating fromthe various points of the diffusion surface. Impulse response of the simple optical system canbe determined as Fraunhofer diffraction at the openings of the diaphragm D. For the complexamplitude in the imaging plane we have [25]

a10(ξ, η) =1

λ2pq

∫∫ ∞

−∞P (x, y) exp[−i2π(ξx− ηy)]dxdy , (3.87)

where λ is the wavelength of light, p, q are the object and image distances, respectively, P (x, y)is aperture function, designation of the Cartesian coordinates is clear from Fig. 3.22. In Eq. (3.87)ξ, η are spatial frequencies defined as

ξ =x1 +Mx0

λq, η =

y1 +My0λq

, (3.88)

where M = q/p is image magnification.The function of the double circular aperture is equal to unity in the holes of radius b and

zero outside the area of openings. Taking only one circular aperture away from the optical axisby d, for the complex amplitude of the diffraction pattern in the image plane we get the knownexpression

a11 =1

λ2pq

b

%J1(2π%b) exp(i2πdξ) , (3.89)

where J1 is Bessel function of the first type and the first order, and

%2 = ξ2 + η2 . (3.90)

The last exponential term in the expression Eq. (3.89) is so-called shifting property of thepositioning the diffraction aperture outside the optical axis of the lens. In the same way weexpress the amplitude distribution for the second opening and the resulting pattern, due to mutualinterference, is the sum of two expressions

a12 =1

λ2pq

b

%J1(2π%b)[exp(i2πdξ) + exp(−i2πdξ)] . (3.91)

The diffraction grating whose amplitude transmittance can be expressed as

tG = C0 + C1 exp(i4πdξ) + C1 exp(−i4πdξ) , (3.92)

where c0, c1 are constants, is placed in this plane. The complex aplitude a12 is transferred to a13

after passing the grating

a13(ξ, η) = a12tG = C0C exp(i2πdξ) + C1C exp(i2πdξ) ,+ C0C exp(−i2πdξ) + C1C exp(−i2πdξ) ,+ C0C exp(i6πdξ) + C1C exp(−i6πdξ) ,

(3.93)


Double-aperture

Lens

Grating

Fig. 3.23. Difraction of the interference pattern on grating placed in the image plane of the imaging lenswith double-aperture diaphragm.

where all the constants of Eq. (3.91) are included in C. As can be seen, the expression Eq. (3.93)comprises 6 terms, and thus the grid will decompose 6 wavefronts. The wavefront described bythe first and the second terms are propagating in the same direction, so it is with the pair of thirdand fourth term (see Fig 3.23).

The objective lens, which displays an image of the subject on the CCD detector, was focusedso that the object plane agreed to the plane of the diffraction grating. Since the different wave-fronts propagating behind the grating are angularly splitted, the objective lens captures only thepair corresponding to the first or the second pair of members in the expression (3.93), respec-tively. Therefore, the following distribution of the intensity appears on the CCD detector

a∗13a13 = (C0C + C1C) exp(i4πdξ) . (3.94)

The intensities of the records in double exposure measurement are summed

I = (C0C + C1C)(

exp[i4πdλq

(x1 +Mx01)]

+ exp[i4πdλq

(x1 +Mx02)])

, (3.95)

where we used the fact that the displacement u = x01 − x02 is very small compared to thecoordinate of the point in the first x01 and second exposure x02. After mathematical treatmentand rewriting to the trigonometric notation, the expression for the interpretation of interferencepattern is obtained

I = (C0C + C1C)2 cos[4πdλq

(x1 +Mx0)]

cos22πdλq

u . (3.96)

The last member of this term expresses searched distribution of the interference fringes, depend-ing on the displacement u. From the conditions for interference maxima follows

u = Nλp

2d, (3.97)

where N is order of an interference fringe.On the basis of that expression, we can assign a value of in-plane displacement in the x

direction to each of interference fringe. Displacements in the y direction are determined thesame way except that the line joining the centers of the diaphragm’s openings is oriented in yaxis.


Fig. 3.24. Double-aperture interferogram on the model rotated around the optical axis. Interference fringesare the isolines of constant displacement component perpendicular to the fringes. The weak contrast iscaused by the principle of CCD detection.

3.4.3 Experimental realization

The optical interferometer assembly with electronic records was created by using conventionaloptical elements: He-Ne laser, collimator illuminating the object, imaging lens (together withthe double-aperture diaphragm represents the principal element of the setup). Focal length ofthe lens was 140 mm and its best numerical aperture was 1/2. The diameters of the diaphragmopenings were 5 mm. For electronic recording and image processing the optoelectronicsystemTELEMET 2 produced by Tesla Piest’any was used. The CCD camera PTK 0384 was connectedto the microcomputer MHB 8080A. The system was equiped with the image record and process-ing software.

The field of surface displacements was simulated by rotating the whole object around theoptical axis. In this case, the isolines of equal displacements in one direction will represent thesystem of fringes parallel to the connection of centers of the diaphragm openings. The exampleof interferogram is in Fig. 3.24. The object rotation was measured micromechanically and byusing this value the validity of Eq. (3.96) was confirmed.

Contrast of the interferogram in Fig. 3.24 is relatively low. This is the common feature ofthe methods utilizing electronic image record, which is determined by the recording principle.Therefore, the additional processing needs to use different electronic or digital image filters.System TELEMET was not able to filter noise or enhance contrast, that is the contrast of thefinal image is weak. However, from the metrological point of view, such a record is definitelyusable.

3.5 Diffraction of light on the surface microroughness

The roughness of the surface, created by natural or artificial way is an important surface propertyand has become increasingly important. The surface texture is a key factor affecting the func-tioning and reliability of a manufactured component. As a rule, the difficulty with traditionalmethods is that they attempt to detect only surface roughness, by which it is meaning surfaceheight variation, whereas surface texture includes many complex and interrelated surface char-


acteristics.In recent years, topographical feature and related analysis technology of engineering sur-

faces have gradually become an important research aspect of machinery and microelectronicsengineering. A typical engineering surface consists of a series of spatial frequencies. The highfrequency, short wavelength components are linked to roughness, the medium frequency to wavi-ness, and the low to form error (basic profile). It is easy to understand, that different manufactur-ing process generates different wavelength feature.

The method of objective judgments and evaluation of the most important component of sur-face structure, roughness, has a long history. At first, a sinusoidal model of unevenness was used,when a chosen quantity was used as an effective (root-mean-square) parameter. Nowadays, themain standard quantities of surface roughness characterization are the arithmetical mean devia-tion Ra and the root-mean-square deviation Rq.

Ra is the mean arithmetical value of absolute deviations of profile within the limits of sam-pling length:

Ra =1l

∫ l

0

|y(x)|dx or Ra ≈1n

n∑i=1

|y(xi)| , (3.98)

where x is the abscissa of the profile subtracted on the mean line, y(x) – the function describingthe profile, y(xi) – the coordinates of n points of surface profile within the sampling length,i = 1, 2, 3 . . . n, l – the sampling length, n – the number of points of surface profile within thesampling length.

At the same time the value Ra represents the central statistical moment of the 1st order,µ1 = Ra. The parameter is geometrically interpreted by the height of the rectangle constructedon the mean line that has the same area as unevenness of profile closed by mean line. The factthat the values of the arithmetic mean deviation of profile Ra presented in this way does notenable us to determine the shape of surface profile itself; this remains the problem of the wholeconception. It concerns namely the height fluctuation of the high-frequency components of theprofile structure.

Rq is a quadratic analogy to the parameter Ra and is defined by prescription:

Rq =

√1l

∫ l

0

y2(x) dx or Rq =

√√√√ 1n

n∑i=1

y2(xi) . (3.99)

The root mean square deviation of the assessed profile Rq is at the same time a standard de-viation (root-mean-square deviation) of coordinated different points of surface profile. It resultsfrom a calculation of the central moment of the 2nd order:

µ2 =∫ ∞

−∞(y −my)2 f(y) dy = Dy = σ2; my =

∫ ∞

−∞yf(y) dy , (3.100)

where my is the mean value of a stochastic quantity. The usual ratio of the above determinedcharacteristics,Rq/Ra, falls within the interval from 1.1 to 1.5 and corresponds to changes in thedisintegration mechanism resulting from various technologies and it also indicates these changes.

From the viewpoint of topography the same objection as for the above mentioned parameterRa is analogically valid for parameterRq. The root-mean-square deviation of profileRq acquires


significance, however, if the profilometry of the surface is carried out optically. It is assigned bythe fact that the majority of optical signals are based on detection of intensity, which is a quadraticvalue of its amplitude.

Non-standard parameters of surface profile serve to provide a more complex view of topo-graphical structure and thus integrally complement data prescribed by standard. The arithmeticmean wavelength of the profile λa, which is a 2π multiple of the ratio of the arithmetic meandeviation of the profile Ra and the mean value of the profile slope:

λa = 2πRa

∆a, (3.101)

where ∆a is the mean angle of the slope of unevenness.The autocorrelation function Ryy expresses the degree of periodicity, or rather the random-

ness of the profile, and it is defined by the relationship:

Ryy(x) =1

N − x

N−x∑i=1

(yi −my)xi(yi −my)(xi − x) . (3.102)

The value Ryy(0) represents the dispersion Ryy(0) = Dy . For the evaluation of the properprofile curve of the surface itself, the so-called standard autocorrelation function is used:

ryy =Ryy(x)Dy

. (3.103)

For many years, information about the surface microprofile has been acquired by mechanicalcontacting of the surface with a sharp tip, whose tip curvature is such, that it can penetrate thedetailed profile geometry.

The light scattering is a sensitive function of its roughness. Part of the light is diffusely re-flected (scattered) into directions different from the specular reflection direction for an ideallysmooth surface. Roughness features, that produce light scattering, are typically separated bydistances from hundreds of nanometers to fractions of millimetres. Larger separations of surfacefeatures, which are the so called surface waviness, contribute only to the near-angle scattering,and its separation on the background of large angle scattering diagram is troublesome, or evenimpossible. On the contrary, microroughness generally scatters light into very large angles anddecreases the amount of light that is detected by an optical sensor. From this point of view, gen-erally, the surface feature heights can be classified according to their ratio with the light wave-length λ in the visible region. In principle, such measuring instruments are limited to measureheights less than half of light wavelength, that is even nanometer scale heights can be evaluated.Such surfaces are conveniently inspected with well established techniques, based conceptuallyon the measurement of total integrated scattering or angle-resolved scattering. However, theseapproaches, based on scalar diffraction theory, do not take into account the lateral texture of thesurface and belong to integrated principles of operation, where only integral information aboutsurface heights variations, generally expressed as the root-mean-square roughness, is obtained.In addition, the interpretation of the results measured needs certain assumptions about the natureof the surface profile, for example, besides the condition rms � λ, lateral dimensions of the sur-face feature must be� λ and the surface structure has to be isotropic or unidirectional. Probablythe most stringent limitation is the demand for Gaussian statistics of the measured surface struc-ture, hence the “natural” surfaces as a rule fulfill these conditions. As the most of the artificially


Light source

Detector

h

x

β

θ

Fig. 3.25. Diffraction of light on the surface roughness. β is the angle of incidence, θ is the detection angle.The dashed-line scattering diagram shows the random character of light reflection from rough microstruc-tured surface.

formed surfaces are inherently deterministic, in engineering practice, correction factors must beused to obtain the true values of surface rms.

The light scattering method shall be used in a variant where the more complete informationabout the reflected light distribution is recorded. A theoretical derivation based on Huygens-Fresnel principle, Fraunhofer approximation, and Wiener-Khinchine theorem shows that theFourier transform of a scattered field is approximately proportional to the autocorrelation func-tion of the surface profile for optically smooth surfaces. This means that the important parameter,surface correlation length, may be simply evaluated.

When the light wave is incident on a solid surface, it is reflected either specularly or diffuselyor both. Reflection is specular when the angle of reflection is equal to the angle of incidencewhich is attribute of mirror-like smooth surfaces. Reflection is diffuse when the energy of re-flected wave is scattered into half space. Generally, the roughness of surfaces covers a widerange of both lateral and heights dimensions from fine grained to rough waved. As roughnessincreases, the intensity of specular beam decreases while the diffracted radiation increases inintensity and becomes more diffuse. The relationship between the light wavelength and the sur-face roughness affects the physics of reflection. The classification of surfaces by their prevailingsurface spatial wavelengths (which means the characteristic lateral as well as height dimensions)can be done.

We are inspecting the surface with a profile described by the function h(x, y) ∈ 〈hmin, hmax〉.The mean value of h(x, y) is h0 and the statistical quantity σ is the square-root of variance orin other words standard deviation. For the sake of simplicity we will assume only 1D situation.Illuminating the rough surface by a plane wave as it is sketched in Fig. 3.25 the phase of the lightwave ϕ(x) due to roughness profile can be expressed as follows [43]:

ϕ(x) =2πλ

(1 + cosβ)h(x) , (3.104)

where λ is the wavelength of light and β is the angle of incidence where also the assumptionabout both sufficiently smooth surface and small surface slopes was applied. Let us consider


aI as the amplitude of planar incidence wave. Provided that the illuminated area is in lateraldimension large enough, the complex amplitude of the light wave immediately after its reflectionfrom the surface can be expressed as a complex function

aR(x) = aIr exp[i2πλ

(1 + cosβ)h(x)], (3.105)

where r is the average reflectivity of the surface. We define microroughness of the surface as aroughness profile with height variations smaller or compared with wavelength of light. In realitythe variations do not exceed several µm. In such a case the phase factor in Eq. (3.105) can beexpanded into series

exp[i2πλ

(1 + cosβ)h(x)]≈ 1 + i

2πλ

(1 + cosβ)h(x) . (3.106)

As known [43] in the position far enough from the observed surface, the complex light fieldcan be expressed as a Fourier transform of the amplitude aR(x)

A(ξ) =∫ ∞

−∞aIr[1 + i

2πλ

(1 + cosβ)h(x)]

exp(−i 2πxξ) dx , (3.107)

where ξ is the spatial frequency. Spatial frequency is a magnitude related to the lateral coordinatein the observation plane normal to the direction of mirror-like reflection from the surface

ξ =u

Lλ=

sin θλ

. (3.108)

In this relationship L is the distance of observation plane and θ is the observation angle from thegrazing direction.

The integral of Eq. (3.107) can be simply divided into two components, where the formercomponent presents zero diffraction order and characterizes in fact an aperture function of beamilluminated area. In the case of finite sized specimen (or more precisely, finite illuminated area)this component could define diffraction on such sized obstacle. On the contrary, when the lat-eral dimensions of illuminated area are much larger compared to the light wavelengths or meanfeature of the surface roughness it presents only intensity contribution into the grazing angle di-rection. In the case of circular illuminated area Airy’s disc distributionit can be written for theintensity

I(%) = C

(d

2

)2 2J1(πd%)d%

, (3.109)

where %2 = ξ2 + η2 and d is the diameter of the diffraction aperture and J1 is Bessel functionof the first kind, order one. In practice, as a rule, the inspected surface area is illuminated byTEM00 laser beam. The energy distribution in this case is characteristic by radially transversalGaussian distribution

I(%) = I0

[W0

W (z)

]2exp

[− 2%2

W 2(z)

], (3.110)


where I0 is the maximum irradiance andW (z) the radius at the I0/e2 point. The beam waistW0

and far-field half-angle divergence is a constant which defines “diffraction limited” minimumsize of beam radius

W0 ≈8λfπD

, (3.111)

where f/D is F-number of the focusing lens.Nevertheless, while the aperture function aI may not be taken into account when dealing

with the analysis of surface roughness parameters, it defines the statistical average of the sizeof individual speckle in diffraction halo. Consequently, we are interested in the second part ofintegral (3.107)

A(ξ) = C(1 + cosβ)∫ ∞

−∞h(x) exp(−i 2πxξ) dx = c(1 + cosβ)H(ξ) , (3.112)

where C is the arbitrary constant and H(ξ) is the Fourier transform of the function h(x). Whendealing with surface roughness, statistical averaging needs the dimensions of the illuminated areato be much larger then the dimensions of the surface grains, that is why we have omitted the firstterm of Eq. (3.107). Light intensity distribution around the direction of mirror-like reflection,which is diffraction halo on the observation plane, is expressed as a square of complex amplitudeabsolute value

I(ξ) = A∗(ξ)A(ξ) = |C(1 + cosβ)H(ξ)|2 . (3.113)

Usually, |H(ξ)|2 is denoted as the power spectral density. Eq. (3.113) indicates that a scat-tered far field distribution is proportional to the power spectral density function of the light waveimmediately in front of surface. According to Wiener-Khintchine theorem, the autocorrelationfunction cf(p, q) of the complex amplitude aR(x) is the inverse Fourier transform of the powerspectral density function |H(ξ)|2. Quantities p, q are the lag lengths of the autocorrelation func-tion. Thus, the Fourier transform of a scattered far field maps the autocorrelation function ofthe complex amlitude immediately after reflection on surface. Now we will consider anotherassumption with regard to statistical properties of the surface profile. Let the function of heightvariations h(x) have its autocorrelation function in Gaussian form:

∫ ∞

−∞h∗(x′)h(x′ + x) dx′ = K exp

(−x2/l2c

), (3.114)

where lc is the correlation length and K is the real constant. We have searched the Fouriertransform of the autocorrelation function∫ ∞

−∞K exp

(−x2/l2c

)exp(−2πixξ) dx = Klc exp(π2l2cξ

2) = |H(ξ)|2 . (3.115)

Now, by substitution of Eq. (3.115) and Eq. (3.112) into Eq. (3.113), we obtain an analyticalexpression for scattered light intensity in the direction of observation θ:

I(θ) ∼ lcλ

(1 + cosβ)2 exp[−(πlc/λ)2 sin2 θ

]. (3.116)


Rel

ativ

e in

tens

ity I(θ

) a.

u.

Observation angle θ

ℓc /λ = 1ℓc /λ = 4

−60° −40° −20° 0° 20° 40° 60°0

2

4

6

8

10

12

14

16

Fig. 3.26. Angle distribution of light intensity by diffraction on rough surface with Gaussian statistics ofmicrorougness with correlation length lc. As expected, the larger values of lc compared to the wavelengthλ, the narrower is the intensity distribution.

Provided that the angle of observation is small i.e. θ � 1 and sin θ .= θ we have the simplifiedrelationship

I(θ) ∼ lcλ

(1 + cosβ)2 exp[−(πlc/λ)2θ2

]. (3.117)

As seen, an important parameter of this function is the ratio lc/λ. The graphic representationof the function Eq. (3.117) for two different values of this parameter and β = 0 is shown inFig. 3.26. As it can be seen, for large correlation lengths bell-like curve is quickly narrowed andpeak tends to clear mirror-like reflection. This tendency is also illustrated by scattering diagramwhich is in fact parametrical formulation of the function Eq. (3.117) in polar coordinates andrepresents geometrical interpretation of graphs in Fig. 3.26. For better view in this Figure theangle of illumination/mirror-like reflection was chosen β = 45 ◦.

The assumption of Gaussian form of autocorrelation function follows from the fact, thatautocorrelation function of “proper” random function shows expressive central maximum and inits vicinity the Gaussian function can be used as the first approximation. Then, the correlationlength lc is defined as a value on x-axis, where the autocorrelation function has a value of 1/e ofits maximum in the central point x = 0

cf = exp

(−R2

q

l2c

). (3.118)

Thus, the horizontal property of one-dimensional surface profile may be characterized by surfacecorrelation length which is a characteristic feature size of surface profile for in-plane dimensions.

Assuming the surface height distribution as a Gaussian random process the standard deviationof roughness Rq as defined by Eq. (3.99) can be determined, too

ψ(h) =1

Rq

√2π

exp[− (h− h0)2

2R2q

], (3.119)


0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

Fig. 3.27. Scattering diagram of the diffracted light from rough surface with Gaussian statistics of micror-ougness with coherence lengths as in Fig. 3.26.

where ψ(h) is probability density function. For the sake of simplicity, we consider one of bothlateral directions. Using relation for the probability density function

ψ(h) =dxdh

, (3.120)

the variables of the integral in Eq. (3.112) can be changed

aS = C

∫h

ψ(h) exp(− i4πλh

)dh . (3.121)

Now, the amount of the light scattered from the rough surface to the direction of mirror-likereflection can be determined. Then, the values of spatial frequencies approach to zero for boththe perpendicular illumination as well as for oblique illumination.

The apmlitude of the light reflected from the ideally smooth and flat surface h(x, y) = 0 is aconstant

a0 = C

∫h

ψ(h)dh = C (3.122)

due to unity of the probability integral Eq. (3.120). Then, in the direction of mirror-like reflection,the ratio of the amplitudes of the light scattered on the diffuse surface and on the mirror-smoothedsurface is the integral of Eq. (3.121). After the substitution of the probability density functionEq. (3.119) we have

R =ISIO

= exp(−4π2

λ2R2

q

), (3.123)

where IS, and IO are the intensity of integral value of the light scattered on the surface roughnessand the light reflected in the centre of diffraction halo. This expression can be used for evaluatingthe value Rq by simple measurement of total amount of the light scattered in diffraction diagramand the part from its central region.


It has to be noted that surface profiles having the same variance Rq may have quite differentcorrelation lengths lc. This property of the profile is expressed in the shape of the autocorrelationfunction by stronger or weaker correlation moments between the profile ordinates.

Eq. (3.123) is the relation between the roughness of the metallic surface and the intensityin the centre of diffraction halo. Since the stress induced roughness is related to the value ofeffective strains, the ratio R can be used as a quantitative measure for the assessment of theplastic strains/stresses.

3.6 Evaluation of elasto-plastic stresses by surface spectrum analysis

The standard procedure of the measurement of elasto-plastic stresses in metals based on straingages or moire often has a week points. This is due to technical difficulties with the large strainmeasurement, more complicated material response and also anisotropy of plastic behaviour. It isknown that increased plastic deformation of metals is followed by corresponding enhancementof the surface roughness. A number of authors have tried to give quantitative interpretation ofthis effect. Yamaguchi and Mellor [44] and Lee et al. [45] have shown the dependence of theinduced surface roughness on the effective strain. Hence, by measurement of surface roughnesschanges the plastic zone contours could be identified. Moreover, Azuchima and Miyagawa [46]and Dai and Chiang [47] have established proportionality between plastic strains and the widthof intensity spectrum of the light scattered from induced surface roughness.

Most of authors refer to applications of the method of correlation analysis of the changes inthe diffraction microstructure of light field and quantities such as spectrum width, contrast, corre-lation coefficient or amount of widely scattered light have been used to quantify this effect. Wellestablished is the relation effective plastic strain vs. intensity level of the speckle image relativeto that of undeformed surface or speckle pattern decorrelation. [45, 48] A linear relationship forsurface roughness and effective strains was observed when the von Mises yield criterion in theelasto-plasticity region was used.

The measurement of plastic deformation and plastic zone around a sharp notch or a crack inthe metal is required not only to study the mechanism of fracture but, also, for design and failureanalysis of a machine. The change in microstructural morphology of the surface often indicatesthe presence of irreversible plastic strains. As the elasto-plastic stress state around the crack tip ina steel material can be numerically computed on the basis of known material loading diagrams,the experimental results of the optical method of far speckle field observation can be confrontedfor. However, up to now a little effort has been done to explain the plastic strain/stress orthogonalcomponents distribution at defined stress state.

3.6.1 Experimental results

The optical diffraction evaluation was used to measure the samples of notched steel beams. Thecrack, whose plastic zone at the tip was to be measured, was simulated by a 0.2 mm wide and17.5 mm long saw cut in the middle of the span. The beam with dimensions of 180 × 35 ×3.6 mm3 was made of high-strength, stainless steel. The observed plastic area at the position ofcrack tip was created by applying of load up to 3 000N by three-point bending with a 140 mmspan of force action. The material parameters were obtained by a standard tensile test. Elasto-


Sample Lens Focal plane CCD plane

Beamstop

fp q

Fig. 3.28. Optical scheme of measurement of diffraction properties of microstructured surface. The beam-stop serves as spatial filter eliminating spectacularly reflected light. Only diffracted and scattered rays areimaged by the lens on CCD.

plastic behaviour of the material is fitted by power law strain hardening constitutive model

ε0ij =32ε0

σsij , ε0 = εr(σ/σr)n , (3.124)

where

σ =

√32sijsij ,

ε0 =

√32ε0ijε

0ij ,

sij = σij − δijσkk

3.

In the above sij is the deviatoric part of the stress, σij is the reference strain at some referencestress level σr and n is the index of power law (Ramberg-Osgood). The values εr (for a givenvalue of σr) and n are 0.015 and 9.1, respectively. Other basic material parameters are: Youngmodulus E = 2.08× 105 MPa, Poisson ratio ν = 0.3 and yield stress σr = 280MPa.

The important condition of the correct results on measured roughness induced by elasto-plastic strains is a metallic surface finished by grinding and polishing before loading of the sam-ple. The polishing does not require to be too high; in the case of rolled surfaces, a moderatemirror-like surface is sufficient.

In measurement of diffraction properties of the studied surface microstructure the specimenwas illuminated nearly perpendicularly to the rough surface with a laser beam in a simple schemeof optical filtration (Fig. 3.28).

The monochromatic light of the He-Ne CW laser with an output power of 15 mW was colli-mated into the diameter of Ø = 20mm and directed onto area around crack tip. Long focal dis-tance lens projects the surface image into the CCD chip of digital camera (Cannon EOS 300D).The area of plastic zone is identified by inserting the iris diaphragm as a low pass binary filterinto the focal plane of the filtering lens. Figure 3.29 illustrates a filtered image of the area wherethe higher spatial frequencies are blocked. Identification of the zone with plastic deformations iswell-marked, nevertheless, the interpretation of the zone boundaries is somewhat complicated.


Fig. 3.29. Filtered image of the plastic zone near the notch in steel polished beam with the higher spatialfrequencies blocked. The plastic zone aroun the notch tip can be clearly identified due to deformationinduced local roughening of the polished metal surface [49].

0.95 4 MPa

0.90 13 MPa

0.80 53 MPa

0.60 124 MPa

0.20 227 MPa

0 2 4 6 8 mm−2−4−6−8

0

2

4

6

8

mm

(a)

300 MPa

3 MPa

15 MPa

40 MPa

90 MPa

240 MPa

140 MPa

0 2 4 6 8 mm−2−4−6−8

0

2

4

6

8

mm

(b)

Fig. 3.30. Comparing the experimentally obtained values of principal stresses in plastic zone around thecrack tip to the simulation by ANSYS [49].

The flatness of the mirror-like area is often affected by mutual inclination of parts of thisarea and the optical filtration makes difficulties. Then the amount of the scattered light can bequantified separatelly for each part of the plastic zone area by scanning the specimen point bypoint with narrow beam. Long focal lens concentrated the laser beam into the circular spot of 0.3to 0.5 mm where it undergoes scattering on the surface. As the evaluation procedure is based onEq. (3.123), only the intensity in the centre of diffraction halo was measured directly by fotodiodedetector. The read value of intensity was then normalized to the value of intensity observed onthe reflection from the part of beam surface not damaged by plastic stress. To separate specularlyreflected light, the fotodiode effective area was screened by a diaphragm opening of Ø = 4.0mmin diameter which is diffraction angle of about 0.5◦. The rays passed through are the rays withinthe diffraction limit for appropriate distance of observation.

Fig. 3.30 shows the curves with equal amount of scattered light from the area within theplastic zone around the crack tip. The curves plotted were obtained as fits of the measured


Am

ount

BofBd

ispe

rsed

Bligh

tB%

0

10

20

30

40

50

60

70

80

90

100

−10 −8 −6 −4 −2 0 2 4 6 8 10

−Y +Y mm

AA'BB'CC'DD'

Fig. 3.31. Amount of dispersed light in several horizontal sections of the plastic zone.

points by the least squares method for the amounts of the dispersed light 95 %, 90 %, 80 %, 60 %and 20 % of the value specularly reflected from undamaged mirror-like area. Provided that weassume Gaussian height distribution of the roughness profile induced by plastic stress, the plottedcurves present equal values of Rq (see Eq. (3.123)). In the figure the smooth character of the iso-curves is clearly visible in spite of some statistical dispersion of the measured data points. This isalso demonstration of the obvious relation between the acting stress and the measure of surfacedamaging.

In order to verify the plastic strain determination method, a numerical simulation of the plas-tic stresses distribution around crack tip was performed by ANSYS SW [50]. In Fig. 3.30bthe computed values of von Misses stresses are drawn. The results can be directly comparedwith those obtained on a calibration specimen for the evaluation of strains. In the calibrationexperiment, steel specimens were uniaxially loaded on a testing machine to the level of plasticdeformation and the scattered field was recorded step by step. The actual strains were measuredby testing machine. The acquired diffraction response of the rough specimen surfaces were cou-pled with the values of one-axis tension strains which can be interpreted as von Mises effectivestrains. It has to be noted that the numerical analyses were performed as a two-dimensional. Forthe thickness of the specimen of 3.6 mm the plane stress conditions are valid to the distance ofabout 1.0 mm to 1.5 mm from the crack tip, the near vicinity is affected by three-dimensionalityof stress distribution. This is visible also from experimentally obtained roughness distribution inthe vicinity of crack tip (Fig. 3.30a) where the shape of effective strain iso-curve is immediatelyrelated through von Mises criterion to 3-D stress state. Inside this zone the butterfly-like shapecharacteristic of plane strain conditions is visible.

Fig. 3.31 gives another representation of the results. The dependences of dispersed lightfound while scanning the elasto-plastic area in several horizontal sections (perpendicularly tocrack path) are shown there. Sudden fall on the curves defines the boundaries of the plastic zonearea. The mapping method was also tested by using the dynamic scanning of the plastic zonearea by uniform motion of the illuminated spot on the rough surface. The signal on light intensity


Fig. 3.32. Z-scanning diffraction halo measurement.

from the centre of diffraction halo was recorded by digital oscilloscope Trace 8086.

3.6.2 Separation of the principal strains

Another way to obtain more detailed information on the diffusion spreading of the sample sur-face is the transversal scanning of the speckle field – diffraction halo, with a suitable opticalpoint-like detector. However, this method abounds in technical difficulties. The photoelectricsignal obtained varies in a large range of intensity values up to 103–105 and the measurementitself is sensitive to non-homogeneities in the diffraction field. These drawbacks can be elimi-nated by scanning the difraction halo in line of sight direction – z-scanning (Fig. 3.32). Planaryphotodetector – two large effective area photodiodes were screened by the diaphragm shaped astwo ±10◦ annular sectors. By moving the photodetector along the optical axis, the light at thediffraction angles from 1◦ to 20◦ may be collected. The range of photoelectric signal is relativelynarrow and the angular intensity distribution is obtained simply and the power spectrum is easilycalculated. The angular distribution of the diffracted light was measured in this manner at severaltenths of points on the specimen surface around the crack tip.

One of the characteristic phenomena observed at this scheme was the expressive directionalanisotropy of the diffraction halo. The diffraction divergency in the x-axis in the crack direc-tion line was lower than that in the y-axis. This arises from the surface roughness defects beingdistinctly elongated in shape with their longer dimensions oriented in the makro-crack line direc-tion. In points out of crack path half-axles of the intensity distribution elliptical shape are inclined(Fig. 3.33). Considering the induced surface roughness governed by Gaussian distribution, thecorrelation lengths can be evaluated using Eq. (3.123).

Determination of the parameters such as speckle contrast, spectrum width or stress-inducedsurface roughness Eq. (3.119) together with the appropriate experimental calibration enableseffective plastic strains to be evaluated quantitatively. The plastic strains evaluation is usuallybased on a condition defined by von Mises for the equivalent stress. However, the complete stressanalysis (at least plane stress or plane strain state) to be carried out needs to know the absolutevalues of the orthogonal principal strain/stress components. The light scattered from the metallic


Fig. 3.33. Elliptical diffraction halo visualises directional anisotropy of surface microroughness. The sur-face rougness defects are elongated in the crack line direction [51].

surface with stress-induced roughness includes also information about the orientations of surfacemicro-unevennesses related to the orientations of stresses. When the surface inside the plasticzone is illuminated by quasi point-like spot, the elliptical shape of the diffraction halo is clearlyvisible. The size and orientation of the ellipse varies point by point on the scanned elastoplasticregion. For simplicity we will assume the grains in a surface layer statistically as sphericallyshaped. These spheroids are deformed into ellipsoidal shape under the action of stresses. Thisis usual assumption in the method of microscopical measurement of the grains shape as wellas in the method of plastic strains evaluation using technique of printed surface circular grids.Actually, empirical experience with metallic surface plastic deformation confirms that the micro-deformations on the base of the order of grain sizes is governed by normal law of distribution.Therefore, the evaluation of elasto-plastic strains has to be realized by detecting the in-planedisplacements that is the microscopic changes in lateral direction.

Gaussian height distribution or standard deviation Rq cannot distinguish the surfaces withdifferent correlation lengths. For isotropic surfaces the correlation length is independent of di-rection along the surface and can be evaluated by Eq. (3.123). Elliptical directional anisotropyof the diffraction halo (see Fig. 3.33) indicates power spectrum of anisotropic surface with thecorrelation length dependent on direction along the surface. A correlation function might thenbe written

cf(x, y) = exp[−(x2

l2x+y2

l2y

)], (3.125)

where lx and ly are the correlation lengths in the x and y directions, respectively. Then the powerspectrum for such surface with a Gaussian correlation function takes the form [52]

|H(ξ, η)|2 =14πR2

qlxly exp(−ξ2l2x/4) exp(−η2l2y/4) . (3.126)

Classic plasticity theory provides the equations with three stresses and three strains in theprincipal directions taking into account von Mises yield criterion and isotropic hardening. Onthe surface free of load, where strains are measured, the stress perpendicular to the surface σ3 =0. At a given point, the directions of the principal normal stresses σ1 and σ2 lie in a planetangential to the surface. For a plane stress condition where σ3 = 0 the equivalent stress σν can


be expressed as follows

σ2ν = σ2

1 + σ22 − σ1σ2 . (3.127)

Since Hooke’s equations represent a linear relationship between stresses and deformations,the equivalent strain εν can be determined directly from the principal strains. Thus, for theequivalent strain

εν =1

1 + ν

(ε21 + ε22 + ε23 − ε1ε3 − ε2ε3

)1/2, (3.128)

where ν is Poisson’s ratio. The strain ε3 can be eliminated on the surface and we obtain

ε3 = − 11− ν

(ε1 + ε2) . (3.129)

Keil and Benning derived the relationship between the equivalent strain εν and the principalstrains ε1, ε2 [53]:

εν =[Nq

(ε21 + ε22

)+Ngε1ε2

]1/2. (3.130)

In this expression Nq and Ng are functions of Poisson’s ratio only, with constant ν the Nq andNg are the material parameters. For constant equivalent strain, εν is constant and Eq. (3.130)represents an ellipse. According to von Mises theory, if equivalent stress exceeds a certain ma-terial dependent limit, elasto-plastic strains occur. Elasto-plastic material response obtained byuniaxial tension test can be formally described in the same way as the linear elastic material.

Laser and optical measurement techniques for testing in microelectronics 191

4 Laser and optical measurement techniques for testing in microelectronics

At the designing process of integrated electronic circuitry, packaging of small structures andnamely the micro/nano-electronic mechanical systems (M/NEMS) device structures the knowl-edge of structural mechanical properties are often essential to the right weighting of all the me-chanical proposal aspects. Nowadays, the development and fabrication of such structures hasbeen realized not only by using conventional well established procedures and known materialparameters. Increasing design and performance demands in the near future will require moreexact and complete information considering both the mechanical and thermo-mechanical ma-terials properties and their mutual interaction in multilayer system. Acquisition of these datais made challenging by the small dimensions involved, mechanical test methods designed forbulk materials are generally not suitable for direct applications to thin film materials, and thinfilm are known to exhibit properties unlike their bulk counterparts. Nonetheless, the mechanicalproperties required for accurate prediction of device performance as well as for proper numeri-cal modeling of the whole system, must be based on reliable, accurate tests. Although many oftools for such a purpose are now established, the fundamentals of mechanical characterizationcontinue to be identified while there is growing need for methods allowing to measure both thematerial and structural element properties.

Laser based metrology in microelectronic technologies owes application in the field mainlyto its specific features, particularly to the contactless and non-invasive way of working. Vari-ous forms of nondestructive, noncontact inspection techniques have been proposed in order tomake possible the evaluating of microelectronic device shapes, their defects and stress induceddeformations, surface roughness, as well as thermal and intrinsic residual stresses, material pa-rameters, and also dynamic performances of MEMS structures. The inherent contactless natureof most of the optical measuring principles predestines many of them to be installed as an appara-tus to perform in-situ measurements inside the vacuum chamber, or in other demanding ambientconditions. To test the wafers, structures and devices, the interferometrical principles are com-monly used, however, the exploitation of photoelectric detecting and CCD camera observationenlarges province to phase shift, white light and scanning interferometry, but also to qualitativeimprovements of classical principles, such as the phase visualization, the Foucault knife or theconfocal focusing. Several methods mentioned above have been applied to inspect the surfaceflatness and roughness of silicon (or GaAs) wafers, to evaluate thin film and membrane resid-ual stresses and to measure both the static and dynamic mechanical characteristics of MEMSmicrocomponents.

4.1 Specular and diffuse-like reflected surface flatness testing

Flatness of upper polished side of silicone (or GaAs based) wafers is one of the vital factors,affecting the reliability of semiconductor device patterns structured by lithography. It can becharacterized as the total indicator reading, or focal plane deviation, describing the distance ofpoints on the top wafer surface to a plane, fitted to this surface. The reasons for such a surfacewaviness can be various. For example, the presence of residual stresses, which is often the caseafter mechanical treatment – polishing of the top wafer surface, can deform the overall shapeof the wafer. Mostly, this global wafer deformation has an anisotropic nature, resulting fromusual orthotropy of main mechanical parameters along the crystallographic axes. The demand


for precise image sharpening at the photolithographic pattern projection needs to know the valuesof surface height variations, particularly when we are dealing with high density circuits. It canbe said that mainly geometrical profile parameters of the virgin polished surface determine thewafer quality with regard to its printability by lithography.

In principle, the measurement of small deviations of specular wafer surfaces can be accom-plished by several optical methods. From the point of view of classical optics, handling withoptically smooth surfaces is a well established metrological task. Therefore, there are manysemiconductor device producers, that are equipped for inspection of wafer flatness and rough-ness with a commercially available testing apparatus.

In spite of that, the commercial, as a rule automated systems are aimed to solve a limitedclass of problems and not always can satisfy the specific and varying requirements, encounteredoften in research laboratories and designing teams. On account of that, the comparative analysishas been performed to evaluate several optical techniques, their possibilities, effectiveness, aswell as limitations.

The surfaces of wafers are often tested by using the arrangement of laser white-field or stan-dard laser interferometry [54]. With regard to wafer measurement, a common drawback of theseinterferometers is their sensitivity, frequently even out of proportion for the testing of relativelystrongly deviating wafer surfaces. Consequently, there is an effort to install oblique incidenceinterferometry [55]. However, the large diameters of the substrates, nowadays usually 150 mmand more, need to have large precise optical elements, such as mirrors, prisms and collimators.

The comparative principle of holographic interferometry eliminates the necessity of preciselarge field of view optics that is why the real-time method of holographic interferometry has beentested. As known, the method is mutual interference of the etalon wave, reconstructed from thehologram, with that reflected off the real wafer surface. The reference hologram wavefront wascreated by recording of the hologram of flat mirror. If necessary, also the reference hologram ofactual wafer surface is recorded and afterwards, the mutual comparison of both the initially notideally flat surface and the surface after forced deformation, respectively, can be done.

Provided that at the hologram reconstruction we have used the wavefront identical to the ref-erence wavefront at the etalon hologram recording, we can describe the intensity of interferencywith object wavefront as follows [42]

I = I0 + 2|a|2 cos(ϕ0 − ϕ1) , (4.1)

where, for simplicity, the equal intensities of both the reconstructed and object waves are as-sumed. In Eq. (4.1), I0 is the light intensity of the zero order diffraction at the reconstruction, |a|is the modulus of complex amplitude, and ϕ0−ϕ1 is the phase difference between the wavefront,reflected from the wafer surface, and the wavefront, reconstructed from the etalon hologram. Ingeneral case, we have the resulting interference field in the form of reconstructed image of theobject, which is covered by macroscopic interference pattern of dark and bright fringes. Argu-ment in Eq. (4.1) characterizes the difference between the etalon and the tested surface. Phase,and consequently path difference depend on the geometry of the optical setup, by which thepaths of both illuminating and reflected rays are specified. In the case of mirror-like reflection,for oblique incidence of parallel rays onto the specimen surface at an angle α we have

w =Nλ

2 cosα, (4.2)


Fig. 4.1. Interference contours of silicon wafer surface deformation.

where w is the deflection of wafer, λ is the wavelength of light and N is the interference order ofthe fringes. In the experiments, the angle of α = 27◦ was adjusted, thus the height difference be-tween two neighboring fringes was 0.355 µm. The holographic record of the flat mirror surfacewas taken on AGFA-Gevaert 10 E 75 photo plates. Precise positioning of the etalon hologramafter its developing was performed precisely to the original position by means of special 3D axesadjustable holder with micrometric screws. Adjusted hologram positioning is considerably sim-plified with regard to mirror-like character of wafer surface, where laser speckles are not presentand lateral shifts are not critical. Also the accurate parallelity of both the collimated beams neednot to be kept, even the spherical wavefronts from point-like pinholes give acceptable condi-tions for etalon hologram adjustment. Fig. 4.1 illustrates one of the holographic interferograms,visualizing deflection contours on the silicon wafer 4-inches in diameter. However, it must benoted that the flatness deviations of a number of wafers tested provided much denser interferencepattern, and the evaluation in such a case became troublesome.

In order to overcome the problem with the range of measuring sensitivity, and also the unprac-tical properties of the experimental setup, the method based on Ronchi tests of optical systemshas been developed. In the optical system, an analyzer grid is positioned exactly into the focalplane of an imaging objective lens. As it is shown below, in such an optical filtering schemethe fringes of slope contours are projected onto the image of object surface. Likewise in theinterferometry, a specular specimen surface is needed. A basic scheme of the optical system isshown in Fig. 4.2. The specimen surface tested is illuminated by a beam of parallel rays, emittedby the point-like source, placed in the focus of a long-focal distance objective. For the sake ofsimplicity of the scheme, the light source on Fig. 4.2 is not drawn.

The wavefront under consideration, reflected from the specimen, is described by the complexamplitude a0(x, y). As it is well-known, the light reflected backward and passed through theobjective lens is transformed in its back focal plane to the Fourier transform A0(ξ, η) of theamplitude a0(x, y). In the focal plane, a rather coarse grid is positioned, which serves as a spatialfilter. This spatial filter is essentially composed of rectangular slits in one dimension parallel tothe y-axis. The field immediately to the right of this spatial filter is thus proportional to

A′0(ξ, η) =12A0(ξ, η)[1 + exp(i2πξ/d)] , (4.3)


Specimen Objective ImageGrid

φ

∂w/∂x

ɑ0(x, y) A0(ξ, η) A'0(ξ, η) ɑI(x, y)

p q

f

x

z

Fig. 4.2. Optical scheme of slope contours measurement.

where phase factors were omitted. In this composition, the second term is the amplitude trans-mittance, where d is the period of lines of the grid. We are searching for the field distribution inthe image plane at a distance q from the lens, then the following convolution integral has to besolved

aI(xI , yI) =ik

2π(g − f)

∫∫ ∞

−∞

12A0(ξ, η)[1 + exp(i2πξ/d)]×

× exp[− ik

2(g − f)((x− ξ)2 + (y − η)2

)]dξdη ,

(4.4)

where k is the wave number. Using Taylor series expansion and after adaptation of Eq. (4.4) wehave

aI(xI , yI) =∫∫ ∞

−∞A0(ξ, η)

[1 +

12

i2πξd

+1

2 · 2!

(i2πξd

)2]×

× exp[− ik

2(g − f)(xξ + yη)

]dξdη .

(4.5)

Also in this equation, the quadratic terms have been omitted. Considering one of the proper-ties of the Fourier transform

(i2πξ)mF (ξ) → f (m)(x) , (4.6)

where f (m) is the m-th order derivative of the function f(x) and F (ξ) is the Fourier transformof f(x). The field in the plane, where the image of the object is projected, can be summarized asfollows

aI(−Mx,−My) =1

4πM2exp[ikM(x2 + y2)/f ]×

×

[a0(x, y) +

λf

4πd∂

∂xa0(x, y) +

14

(λf

2πd

)2∂2

∂x2a0(x, y)

],

(4.7)

where M = f/(q − f) is the image magnification and f and q are the focal distance and theimage plane distance, respectively. In the expression for complex amplitude, the basic terms


from the expansion of spatial filter function present zero, first and second order derivatives ofthe input complex amplitude. The relation between the height function of the reflected surfacew(x, y) and the complex field in the (x, y) plane immediately to the right of the surface is acomplex magnitude

a0(x, y) = exp[i2πw(x, y)/λ] . (4.8)

For quadratic detection of the light intensity distribution, simply the complex amplitude mustbe multiplied by its complex conjugate value

I(−Mx,−My) = a∗I(−Mx,−My)aI(−Mx,−My) . (4.9)

Performing this multiplying, an expression is obtained, where the different derivatives arepresent

I(−Mx,−My) ∼

∣∣∣∣∣1 +K1∂w

∂x+K2

(C1∂2w

∂x2+ C2

(∂w

∂x

)2)∣∣∣∣∣

2

, (4.10)

where the second derivative term has been taken as the last one. K1,K2, C1, C2 are the constantsrelated to the value of λ/d, the last quadratic term can be omitted due to its negligible value.

The influence of different terms on the resulting pattern is related to the ratio of λ/d. Whenthe grid with a large distance between the slits is used, an image of object area is visible asthe zero derivative of the input signal with the system of fringes of the first order derivative,overlapped on the image. Geometrically, the first order derivatives can be regarded as slopecontours of surface deflections. Turning the grid around optical axis, the pattern of slope contourswith regard to orthogonal y-axis can be simply obtained. In addition, by proper choice of thegrid period, even the fringe pattern of slope contours, modulated by the second order derivatives– radii of curvatures – can be realized. The fringe value constant can be obtained simply by usingelementary ray tracing. The rays under consideration, reflected from the specimen, are passingthrough the slits of the grid. These slits are projected onto the object image as a fringe patternof slope contours. Taking into account the law of reflection and thin lens geometrical opticsrelations, we can get the expression for interpretation of fringes

∂w

∂x=

Nx

2νf,

∂w

∂y=

Ny

2νf, (4.11)

where w is the plate deflection and ν is the frequency of the grid.As it can be seen from Eq. (4.11), by proper choice of grid frequency, the sensitivity of the

measurement can be adjusted. Nevertheless, the limitation is obtained when high-frequency gridis used, which is followed by appearing of diffraction effects, as it is explained above. It impliesthe optimum frequency of the grid in a range 1–10 lines per mm at the optical arrangement withthe focal distances of objective lens of about 500–1 000 mm. In order to illustrate the fringepatterns, they are presented in Fig. 4.3. In the figure, there are orthogonal slope contours patternson the same specimen.

For demonstration purpose Fig. 4.4 shows application of more dense linear grid, when alsohigher terms of Eq. (4.10) are considered. The additional system of clearly visible fringespresents second order derivatives curves of the thin plate surface deflections.


Fig. 4.3. Slope contours on the wafer surface visualized by the Ronchi grid.

Fig. 4.4. Second order derivatives of the surface profile deformation on centrally loaded thin plate.

The problem of high sensitivity measurements of the interference contours on the surfaces ofthin plates, which must often be solved by a special arrangement of the optical scheme, can berelatively easily removed by the shearing interferometry. Even large diameters of the field in theshearing interferometer can be implemented in the holographic variant, as described in section2.3.

Mirror-like surfaces of semiconductor wafers or membranes are mostly deviated to the nearlyspherical concave or convex shape and can thus be regarded as a simple optical element. Quasi-spherical mirror element reflects collimated light beam as a concave or convex mirror and createsreal or virtual focus. In order to determine the focal distance or curvature, an autocollimationarrangement was adjusted (see Fig. 4.5). Besides the wafer specimen, the optical scheme is com-posed of only long focal distance lens and the screen moveable along the optical axis. Likewisein optical method of slope contours measurement, the specular surface of the tested object is il-luminated by a collimated beam of parallel rays. The main objective lens oversizes the specimenby the diameter and serves as a collimator with the point-like source placed precisely in the frontfocal point. After the reflection back off the wafer surface, a light beam enters the lens oncemore and the wavefront becomes convergent. If the light is reflected from a plane surface, the


Image planeposition

Lens

Laser

Undeflected

l f

Δ

DeflectedWaferwafer

Fig. 4.5. Optical scheme of wafer curvature measurement based on autocollimation principle.

rays are collected backward into the focal spot. In Fig. 4.5, for the sake of simplicity, the lightis not drawn. In technical realization, the illumination is installed from a side and directed alongthe optical axis by the splitting cube. In the case of either concave or convex shape of specularsurface, the wafer acts as an optical element – spherical mirror – and this shifts the focus ofreflected light along the optical axis either towards the objective or to the opposite direction.

Assuming the specimen as a spherical mirror, the change of focal distance of the returnedwavefront can be expressed by the well known relationship for a system of two centered thinlenses at the mutual distance l

1f

=1f0

+1fs− l

f0fs, (4.12)

where f is the focal distance of two-lens combination and f0 and fs are the focal distances of theobjective and the specimen, respectively. Taking into account R = 2fs, the radius of curvatureof the mirror-like wafer can be written

R = 2[f0

(1 +

f0∆

)− l

], (4.13)

where ∆ is the shift of image (focal plane) of the whole system with respect to the original focalplane.

As it has been already noted, as a rule repeatedly the monocrystal wafers but also even thinmembranes surfaces are deformed ellipsoidally due to materials anisothropy. In this case, theorthogonally unsymmetric surface shape reflects the rays in the form of an astigmatic beamof rectilinear rays. As it is known from basic ray tracing laws, two planes, which contain theshortest and the longest radius of curvature, are perpendicular to each other. The correspondingcurvatures are usually called tangential field curvature 1/Rt and sagittal field curvature 1/Rs.The quantity

1R

=12

(1Rt

+1Rs

)(4.14)

is their arithmetic mean value. Both the radii Rt and Rs can be determined by using the auto-collimation scheme. In this case, the backward astigmatic beam is focused into two focal lines,


Planparallellaser beamillumination

Camera

Central deflectionWafer

Grid

Chip

Fig. 4.6. Optical scheme of shadow moire method.

oriented orthogonally one to the other at the separated focal planes, which by Eq. (4.12) definethe corresponding values of Rt and Rs.

The realization of the optical setup consists of a laser diode chip as a point-like emitter650 nm/10 mW, well corrected telescope doublet f = 1000 mm, Ø = 152 mm and a smallground screen, moveable together with nonius scale on a rail with a mm scale. For better search-ing, the focal spot is observed by a simple magnifier or by using of microscope eyepiece. Theachievable precision of such a focal plane localization is under 1 mm. In spite of its simplicity,the method has proved to be very effective and reliable with the resolving power of radius ofcurvature determination comparable or even better to that of classic interferometry. Moreover,the presence of anisotropy in the wafer deformation is simply discovered and quantitatively de-termined. As for its main characteristics, it can be stated that there are no technical problems todeal with wafers of sizes from 10 mm to 150 mm and more in diameter, the method can be usedto monitor the bulging of large membranes, and moreover, there are also expectations to solvesimply the task of anisotropic thin film stress evaluation with a stress resolution of ±1 MPa andup to 10 GPa range.

In testing of semiconductor substrates or when thin film residual stresses have to be deter-mined, there is a class of problems, where the surface of the substrate scatters the light diffusely.In these conditions, the shadow moire method has been analyzed to acquire the wafers deforma-tion. The primary purpose of the method is to measure the out of plane shape of the plate. Atthe simple optical setup, the reference grid with the frequency of 10–40 lines/mm is contactedimmediately on the tested surface (see Fig. 4.6). Observing this shadow perpendicularly to thesurface, the specimen grating is distorted by the deflections of the surface. When such a distortedimage of the grid is viewed through the same linear reference grating by eye or by CCD camera,moire fringes are created. These fringes represent the surface contours i.e. the surface topol-ogy [56]. Dealing with wafers, the ability to measure small deflections is the basic criterion ofthe applicability of the method. In this meaning, the smallest measurable deflections are definedby diffraction phenomena on the reference grid. As it can be also deduced from the illustrativeexample in Fig. 4.7, taking the surface away from the grid plane in the wafer central part, thecontrast of fringes is decreased due to diffraction, which is also the main limiting factor. The radiiof curvatures of the presented specimens areR = 6m on the left-hand side andR = 71m on the


Fig. 4.7. Shadow moire fringes on the diffuse-like slightly spherical surfaces of wafers.

right-hand side. In any case, the sensitivity of the measurement can be considerably improvedby an intensity interpolation procedure between two neighboring fringes.

4.2 Residual stress measurement of thin films, membranes and MEMS components

Thin films, deposited on substrates, are as a rule in a state of residual mechanical stress. Nanos-tructured thin films of controlled properties such as mechanical, electrical or piezoelectrical, areone of the main aims of nanotechnology. Multilayers and gradient thin films of continouslyvarying mechanical properties are exploited for controlled interphases in multiphase materialsas nanocomposite materials. Mechanical stresses induced in mechanical structures can have adetrimental effect on their mechanical, but also electrical properties. Since the mechanical stresshas a large influence on the stability, flatness and durability, particularly on thin membrane-likestructures, their values have to be analysed together with the monitoring of stress response duringseveral technological processes.

In general, experimental detection and evaluation of the mechanical stress state can be dividedinto two categories:

1. measurement of steady state deformation induced by residual stress in free-standing struc-tural elements

2. measurement of deformation forced to clamped structural elements such as thin plate ormembrane by controllable loading

At the present state there are some experimental-analytical methods intended especially tomeasure residual stress in thin multilayered wafer structures. The residual stress of the layeredstructure can be considered as the sum of two separate components, thermal stress σth and in-trinsic stress σin, which are related by the additive manner

σr = σth + σin . (4.15)

The thermal stress is a function of temperature change during deposition process and themain part of the intrinsic stress is induced by the mismatch of both the crystalline lattices at theinterface of adjoining layers. Such a stress is related to the mutual combination of materials,


conditions of growing process and especially on a diffusion process. The thermal component ofthe clamped-clamped structure can be described by a simple relation

σth =E

1− ν∆α∆T , (4.16)

where E is Young’s modulus, ∆α is the difference of thermal expansion coefficients and ∆Tis the temperature difference. To evaluate thin layer mechanical stress the simple analyticalexpressions are used but precise reliable residual stress distribution evaluation leads to the needof applying right constitutive strain-stress relationship. Moreover, there is also uncertainty inintrinsic stress assessment in crystaline mesh which is not covered by common approximationsof classic continuum mechanics.

The well-known measurement of film stresses of coated wafer or substrate of cantilever beambelongs into the first category where the free deformation induced by stress is measured. Thepresence of residual stress creates the buckling residual deformation in thin plate type compo-nents. Knowing this deformation, thin film stress can be determined through the use of a simpleanalytical expression, Stoney’s relation [57]. The biaxial stress in the film relates to the changein substrate curvature as follows

σf = Mh2

s

6Rhf, (4.17)

where hf is the film thickness, hs is the substrate thickness, M is the biaxial modulus of the sub-strate and R is the induced curvature due to the residual stress. Since thin plate deformation forhomogeneous isotropic material has only a simple spherical shape, experimentally the problemis reduced to the determination of radius of curvature of the buckling. In the field of microelec-tronics monocrystalline materials we are dealing mostly with orthogonal materials, consequentlyanisotropic residual stress distribution is present. Then, on the basic assumptions of thin plateelastic theory, for orthotropic substrate material it can be obtained

∂2w

∂x2=

1Rx

= 6hf

h2s

(1c11

σ1f +ν12c12

σ2f

),

∂2w

∂x2=

1Ry

= 6hf

h2s

(ν12c12

σ1f +1c44

σ2f

),

(4.18)

where w is the deflection of thin plate wafer deformation, hf and hs are the thicknesses of thinlayer and substrate respectively, σ1f , σ2f are thin film layer anisotropic stress components andc11, c12, c44 are anisotropic thin plate – substrate parameters. As follows from Eq. (4.18) todetermine both the anisotropic stress components only curvatures mutually orthogonal of theplate deformation have to be measured.

As it was mentioned above, a variety of optical methods for flatness inspection is possibleto use for curvature measurement. From the point of view of easy realisation, for example, theresidual stresses can be determined by autocollimation measurement with a reasonable precisionof several MPa in a large range of film thicknesses from several nanometers to micrometers.The accuracy of the above described methods is usually satisfactory for reliable determination ofstresses also on the samples as small as 10–20 mm in diameter, even without the installation ofphotoelectric or CCD based detection and/or somekind of signal processing.


Fig. 4.8. Scheme of double-layer structure curving.

Fig. 4.9. Three-layers basic composition of the cantilever [24].

Similar residual stress determination, based on Eq. (4.17), can be realized also on free-standing double layer strips of cantilevered beams. Assuming the cyllindrical-like curving ofthe cantilever in this case, the end of the beam clamped at one of the ends is deflected.

During the process of development of the sophisticated GaAs based microwave power sen-sors with the integrated thermal converter (MTC) device as a heart [58], the investigation ofthermomechanical properties of the device was one of the essential parts of design. The multi-layer heterostructure based design is often complicated. It consists of various materials of eachother different thermomechanical parameters. For optimization of the thermomechanical designusing 2D models, especially for membrane-like structures, is reasonable.

To define the optimum configuration of GaAs based MTC device, three basic types of me-chanical elements has been investigated, the cantilever beam, the doubly clamped bridge andalso microisland element. All the elements have the same or similar GaAs or GaAs/AlGaAsmultilayer membrane structure.

The numerical analysis of the structural element that was carried out required to be supportedby experimental verification and testing of thermomechanical properties.

The characteristic feature of the clamped cantilever beam is its large steady state deformationinduced by thermal stresses created at the process of deposition. The overall cantilever structureis composed of three basic layers – GaAs and metal films Ti/Au (see Fig. 4.8 and Fig 4.9). Withregard to the flat shaped structure, thin plate plane strain model can be used based on the theoryof bi-metal thermostats. Basic formulas for bi-layer strip element thermal deformation werederived e.g. in [59]:

d2w

dx2=

1R

=εf(h1 + h2)

2Θ(D1 +D2), (4.19)


Θ =112

(h2

1

D1+h2

2

D2+

3(h1 + h2)2

D1 +D2

),

D1 =E1h

31

12(1− ν21),

D2 =E2h

32

12(1− ν22),

(4.20)

where w, h1, h2 are the magnitudes of double-layer geometry – deflection, thicknesses of thefirst and the second layer, respectively, E1, E2 and ν1, ν2 are the Young modulus and the Poissonratio, respectively, D1, D2 are the flexural rigidities, R is the radius of curvature. The value ofεf is the free-standing strain, that is for thermal problem

εf = ∆αth∆T , (4.21)

where ∆αth is the difference between coefficients of thermal expansion of the layers and ∆T isthe temperature difference. The free-standing deformation induced by each of layers is additive,i.e. the multilayer system can be evaluated step by step after depositing of each of the layers.

1R

=N∑

i=1

1Ri

. (4.22)

For the specific case if the thickness h1 of the film is much more thin than the thickness of thesubstrate h2, the previous formulas lead to the expression

1R

=6εfh1

h22

, (4.23)

that is for film stress of each of the layer σi Stoney’s formula Eq. (4.17) is obtained.

4.3 Microinterferometry

Dealing with specular surfaces in order to characterize 3D profile of small MEMS structuresand elements, mostly the interference principle is used. Using the coherent laser light, the in-terference between light reflected from the surface and that returned back from a reference flatproduces interference fringes. The resulting fringe pattern is a contour map of the phase differ-ences between the two wavefronts.

The basic optical element of the microscopic laser interferometer setup is light splitting cubeof two prisms. The interference effect is appeared in the air gap between the cantilever surfaceand outlying bottom flat of the splitting cube. That is the mutual interferency is created by theinteraction of two light wavefronts, reflected from object surface and that reflected back frombottom flat of the cube.

The laser beam from He-Ne laser was extended to wide diameter beam of light by usingcollimator (see Fig. 4.10). This basic scheme of interferometry is generally named as Tolanskyarrangement.

In the interferometer the optimal intensity conditions were adjusted by rotating of plane of po-larisation of the laser source. By this way also disturbing secondary reflections can be minimisedand the interference pattern is not blurred by parasitic fringes. In microelectronics elements the


Fig. 4.10. Scheme of the Tolanski microinterferometer.

Fig. 4.11. Steady-state deformation of the multilayer membrane bridge [60].

amount of light reflected from the surface and that of flat reference glass surface not covered by areflection coating are adequate to create contrast interference pattern. The splitting cube is fixedon a two axes adjustable holder. Besides the good flatness of the cube surfaces, the only criticalelement is aberration free collimator objective. The image of object is viewed and magnified ata suitable measure by microscopic objective and built-in CCD camera. Regarding finite size ofthe beamsplitter cube, a microobjective with long working distance has to be used. Large field ofview, easy adaptation of the arrangement to variety of purposes and also the potential possibilityof simultaneous observation of thin film interference pattern on transparent surface covering arethe practical advantages of the device.

Several kinds of the structural elements made of Si, GaAs and GaN based technology ofmembrane-like structures have been observed by using this tool (see e.g. Fig. 4.11).

In Fig. 4.12 from the fringe pattern along the length of cantilever the profiles of curvature aredrawn. As it can be seen, besides the visualisation of steady-state profile after its technologicalforming, the changes generated as a thermal response by acting different feeding power can beinspected, too.


Fig. 4.12. Free-standing deformation of the GaAs based multilayer cantilever thermally loaded [61].

Fig. 4.13. The interference fringes obtained by white light interferometry of the bulging of AlGaN/GaNmembrane 4.2 µm thick and 1.52mm in diameter at pressure 8 kPa. Fringe value is 0.263 µm.

In microinterferometry, there are three basic optical schemes – Michelson, Mirau and Tolan-sky interferometers, depending on the position of reference glass flat in the arrangement. As arule, the optical arrangement is integrally built into the microscopic lens. Each of these types ofgeometry has its own specific features. Michelson type of interferometer is e.g. regularly usedin connection with microscopic objectives of smaller magnification 1× to 5×, larger viewingfield and also larger working distance. An example of deformation contours on the AlGaN/GaNbased membrane bulged by overpressure is shown in Fig. 4.13. The interference pattern wasobtained by white light interferometry using 525 nm LED light wavelength (WLI Bruker Con-tour GT-K1). Low coherence or white light interference microscope is an advanced tool withsome specific advantages over the “conventional” interferometric technique. It is primarily theability to strongly reject light that has undergone scattering outside which gives the generationof speckles when illuminating by coherent light. When a low coherence light is used in inter-


ference microscope, and the microscope objective is moved continously in line-of-sight axis, thecontrast of interference fringes is modulated depending upon the optical path difference. A ba-sic principle is searching for the position of maximum contrast simultaneously for an array ofimage points. Thus, a surface profile can be measured by finding the maximum peak positionof the fringes modulation in a CCD camera. Nevertheless, the position of optimal fringe patterncontrast for low LED based light source can even be found, thus providing the opportunity toobserve immediately the whole field of view as seen in Fig. 4.13.

The task to evaluate the stress of tensioned membrane can be solved by bulging method orby the method of resonant frequency determination. Bulge testing was one of the first techniquesto study the membrane stress. During the test, a uniform pressure is applied to one side of themembrane, properly clamped or supported over its edge.

The bulging method of tensile stress measurement is based on the second order differentialequilibrium equation by which the relationship membrane deflection vs. acting load is described.The deflectionw(x, y) of the stretched membrane under an external pressure load p(x, y) is givenby Poisson’s equation

∆w(x, y) = −p(x, y)τ

, (4.24)

where ∆ is the Laplace operator, x, y are the coordinates of the point of the membrane surfaceunder consideration, p(x, y) is the constant load, perpendicular to the membrane surface and τ isthe tension force per unit length of an arbitrary shaped membrane boundary. Membrane tensilestress is related to the force τ and the membrane thickness h as

σ =τ

h. (4.25)

For small deflections assumption the second order derivatives in Eq. (4.24) can be express by theprincipal radii of curvatures Rx , Ry with respect to the orthogonal coordinates

1Rx

+1Ry

= −pτ. (4.26)

For experimental of the membrane tension evaluation usually these radii of curvatures have to bemeasured.

Experimental evaluation of the initial membrane prestress is performed following Eq. (4.26).In order to distinguish between the intrinsic prestress σ0 and the membrane stress induced by thebulging deformation we get

σ0 + σp = −pR2h

, (4.27)

where σp is the induced tensile stress. The pressure induced stress can be sinply calculated usingHooke’s law and geometrical relations of the bulged membrane deformation

σp =E

1− ν

r2

6R2, (4.28)

where r is the half diameter of the membrane contour.For small sizes membrane like in Fig. 4.13 the tools of microinterferometry can be used, but

for membranes with lateral diameter up to 150 mm the application of autocollimation technique


2Δφ

Collimator

Laser diodeBeamspliter

Cantilever beam

PSD

w

Fig. 4.14. Schematic drawing of the cantilever deflection measurement by PSD.

has been chosen. The deflected surfaces of the membrane in their central region are alwaysnearly spherical, so that the autocollimation focus searching provides directly the informationimmediately necessary to find the tensioning of the membrane. In order to improve the sensitivityof the best focal plane searching, the Fourier diffraction analysis of the near vicinity of focalplane region has been done. In this region, the focal spot in detail visualizes any deviations ofthe membrane surface from an ideal spherical shape and the methodology has served to improveresults to a very good measuring precision. Another advantage of this approach is also directoptical fitting of the mean spherical shape in the presence of local surface deviations. Suchsurface distortions are well observed as the focal spot blurring. The theoretical/experimentalanalysis of the best focus detection has led to a very good stress measurement accuracy achieved,not worse than some tenths of a MPa in the range of membrane tensions 1–20 MPa.

The steady-state deformation induced in double or multi-layer cantilever clamped in one ofits leads to a simple circular shape. Therefore, the curvature can be deduced from inclination ofits free end area. The laser reflectance scheme was arranged to monitor the deflections at the tipof microcantilever of only 340 µm long. A narrow laser beam was focused through the smallcollimator onto the small area specularly reflecting the light backwards to the position sensitivedetector (PSD), see Fig. 4.14. The linear PSD captures the differences in the electrical signalsfrom two opposite ends of the common photodiode substrate. Provided that the characteristic ofPSD is linear, the linear relation exists also between the output signal U and the change in surfaceinclination. At the small slopes of surface the deflection w is related to lateral coordinate x

U = K2dwdx

, (4.29)

where K is the factor of proportionality – sensitivity value. The sensitivity value was calibratedat static conditions as a transfer characteristic by varying the angle of inclination. In the calibra-tion a very good proportionality was obtained in spite of considerable distortion of the reflectedlight spot on the effective PSD area, it implies that the accuracy of measurement is not affectedstrongly by this frequently happened effect. Besides the coupling of the output signal with in-terferometrical deflection measurement, the comparison was made with the analytical solutionof double material thermal deformation. Provided that the rectangular cross section cantilever


is composed of two uniformly thick layers, the radius of curvature of the arc-shaped microbeamshould be

d2w

dx2=

1R(x)

=h∆α∆T

2ΛD, (4.30)

where ∆α, ∆T are mismatches between coefficients of thermal expansion and temperatures, re-spectively, h is the whole thickness of the microbeam, D and Λ are constants related to material.Changing the radius R(x) the variations of inclination angle at the tip of clamped microbeamcan simply be calculated.

4.4 Vibration measurement of multilayer membrane-like microcomponents

As it was mentioned above, measurements of internal or residual stress in microstructures werecommonly realized either by observing the static shape residual deformation or the deformationforced to element. The values of resonant frequencies of the membranes as structural elementsare related predominantly to the tensile stress, acting to keep it flat. Thus, knowing the value ofresonant frequency, the membrane stress can be calculated.

The apparatus of Laser Doppler Vibrometer with pointwise sensing of vibrations was usedfor the detection of vibration movements of both the macro as well as microelement of themembrane-like structural components. The applicability of resonant frequency values obtainedin the ambient air was established experimentally only for stress evaluation on small size mem-branes as a maximum some mm in diameter. Then a proper frequency range for stressed mem-branes is 1–20 kHz. In the case of large membranes, such as e.g. Ø126 mm, the air damping isinaccessibly high and the stress evaluation is impossible. To overcome the problem, a vacuumapparatus was arranged, where the membrane inside the vacuum chamber was excited electrostat-ically and its periodical bulging vibrations were observed by Laser Doppler Vibrometer probingbeam through the optical window. The developed methodology of membrane stress determina-tion has achieved good accuracy in the order of 0.1 MPa.

Small dimensions of the bridges, with lengths varying from 150 µm to 900 µm and withwidth 90 µm, limit the conditions of tiny mechanical loading. Frequently used the electrostaticmethod for mechanical exciting was not appropriate, then the loading through the acousticalcoupling was chosen. Two variants of the acoustic excitation can be applied there – by shortacoustic pulse or by scanning of the appropriate frequency range.

Circularly shaped micro-membranes with diameters of 750, 1 000 and 1 500µm were excitedby acoustic short time pulse generated by mechanical shock. The frequency content of this pulseas a wide band excitation reached the frequencies up to 200–300 kHz. The excitation was carriedout at atmospheric ambient conditions. The measured samples were fixed in a holder and the fo-cused spot from the laser vibrometer head scanned the area of diaphragm point by point. Analogsignal from the vibrometer head Polytec OFV-302 was preprocessed by OFV-2601 controllerand then recorded by LeCroy 808Zi. The intrinsic membrane residual stress was evaluated byiterative procedure comparing experimental frequency value with that of numerically obtainedby 3-D FEM modeling using ANSYS SW tool.

Another way of excitation of mechanical movement by sound can be regarded as very desir-able because of its tenderness and simple handling with both intensity and frequency adjusting.The main difficulty here is the problem with quantifying of the acoustic pressure values. We


Fig. 4.15. Microinterferometrical view of the membrane bridge with coplanar waveguides.

attempted to get over this drawback by calibrating of acoustic pressure emitting on loud speakermembrane by precise measurement of membrane vibration velocities. The method was tested onmicrofabricated GaAs based membrane bridge 790 µm long with the thickness of about 1.0 µm(see Fig. 4.15). This absolute calibration of membrane element bulging is based on the relation-ship between the velocity of longitudinally vibrating air particles induced by speaker and theperiodic pressure changes

p = %cv , (4.31)

where v is the velocity of vibrations, and the composition of air density % and sound velocityc is the so-called wave resistance. At room conditions this value is %c = 415 kg m−2s−1. Inthe nearest neighborhood of the exciting membrane the same value of acoustic pressure can beaccepted. For plane wave of sound the acoustic pressure and acoustic velocity are in phase thatin Eq. (4.31) can be used to calculate the pressure. LDV is an ideal tool to gauge the velocitiesof loud speaker membrane vibration, hence the value of acoustic pressure can be accuratelydetermined precisely.

Another approach how to realize sound excitation with controlled value of acoustic pressurehas been proposed. For this method a small chamber of pistonfon generated periodic harmonicchanges of uniform pressure under the small membrane. Pistofon Metra RFT PF-101 designedfor microphone calibration provides nominal pressure 118 dB sound level at 173 Hz, which isrelated to 15.8 Pa (RMS) of the acoustic pressure. Usually, the range of sound intensity of sometens of Pa is adequate to experiments with microcomponents.

The tension of a tested bridge membrane defines the first resonant mode at the region of 60–100 kHz, thus preventing the resonant excitation in the cavity of pistonfon. On the other handthe expressive bulging oscillations of the bridge are presented at low sound frequencies wherethe pistonfon or loud speaker actuation is very effective. The amplitudes of the oscillations areso high that the movements at hundreds of Hz have a quasi-static nature with negligible inertialeffect. Therefore, the static bulging can be taken into account and the appropriate expressionsfor memrane bulging can be used to calculate the tension knowing the absolute value of actingacoustic pressure.

The schematic drawing of the LDV measurement setup used to perform the measurementsof the membrane like elements is shown in Fig. 4.16. The heterodyne LDV was OFV-303 in-


RC Generator

Preamplifier

LDV HeadPolytec OFV 303

Controller

Loud speaker Membrane

PC

A/D Converter

Fig. 4.16. Experimental setup of the micromembrane element bulging measurement by Lasrer DopplerVibrometer.

Fig. 4.17. Frequency characteristic of the multilayer microcantilever. The vibration of the cantilever wasexcited by sound preassure [61].

terferometric head with controller OFV-2601 made by Polytec. The useful signal from the headamplified and decoded by the controller proceeds in analog form through A/D converter Advan-tech PCL-818 to PC.

The maximum deflections measured varied from about 30 nm to 100 nm which values corre-sponded to 100–120 dB of acoustic pressure, accordingly the evaluated tension was found to bein the range 10–25 MPa. These stress values were approved independently by numerical com-putation using MEMCAD Coventor Ware simulator but also by residual stress state monitoringduring the deposition and etching procedures.

Having a possibility to actuate simply such a double-layer system, the measurement of res-onant frequency curve was offered by itself. The plotted resonant characteristic is shown inFig. 4.17. As seen, the peak is remarkably sharp which fact leads to small damping factor. The


reduced damping factor γ = 4.5×10−3 was obtained by using the half power bandwidth method.However, as it must be noticed, the experiment with the LDV detector was realized in the ambi-ent air thus lowering the vibrations. Although the air damping effect tends to be negligible withsmallest vibrating objects sizes, precision determination of the damping value needs the sampleto be installed in vacuum.

Accordingly, the well defined ambient conditions are necessary when the effect of surfacestress induced bending or thermal mismatch on resonant frequency shift is analyzed. We studiedthe same effect in connection with curved static residual deformation but the results were notconvincing.

On the contrary, the effect of resonant frequency shift was detected on 3-inch silicon wafersdeposited with 1 µm film of amorphous silicon. Before and after the deposition, the wafer defor-mation (radius of curvature) was optically measured in order to determine the residual stress. Thefrequency characteristic of the wafers was evaluated by LDV in the same setup as it is drawn inFig. 4.16. From the face side – polish surface of the specimen, the membrane of the loud speakeris acoustically coupled on the disc of wafer. For the tiny and thin walled elements of siliconwafers there is a key factor to secure the mechanical vibration to be free of any undesirable in-fluences. In the experiment the wafer disc was hanged on two tiny textile fibers with orientationof disc surfaces vertically. The fixation with flexible hanging realizes reliably free-free bound-ary conditions, any other fixation either clamped or bending is not experimentally reproducibleenough to avoid uncertainties. With regard to that aspect it is better to use the reflection of theprobing laser beam from the back side diffusely scattering the light. In the case of using polishedside reflection the small reflection angle fluctuations of the hanged specimen can cause problemswith the loss of signal outside of LDV head objective. The resonant frequencies measured ontwo wafers are listed in Tab. 4.1 where also an analytical calculation values are given.

4.4.1 Visualization of different mode shapes of thin membranes

In addition to the experiments on large 6-inch in diameter silicon stencil masks membranes thesmall silicon membranes with square-shaped geometry were analyzed on the membrane tensionstresses. The specimens with the thickness of about 3 µm and dimensions 10 × 10 mm2 or5× 5 mm2 respectively, were used.

To test the small membranes two concepts were designed, the former based on LDV de-tection of vibrating membranes deflections and the latter used the autocollimation scheme andphotoelectrical signal acquisition.

Tab. 4.1. Experimentally measured shifts of resonant frequencies affected by film stresses.

Without film With filmResonant frequency Hz Experimental Analytical ExperimentalSample 1 Mode 1 1 105.5 1 103 1 113

Mode 2 2 520.7 2 513 2 524

Sample 2 Mode 1 1 082.7 1 082 1 086Mode 2 2 471 2 465 2 471


In principle, no essential difference in measuring setup of membranes when using laserDoppler vibrometer was carried out (see Fig. 4.16). Though, handling with such a delicate struc-ture in atmosphere, care must be taken with regard to adding mass of membrane neighboringair gas. This means that the acquired resonant frequencies in the air ambient have to be cor-rected for air damping to obtain their true values. Mostly the empirical expressions are used forsuch corrections more or less successfully. Analysing a number of our mesurements performed,the frequency of the first natural mode has to be corrected by a factor of fvac/fair = 1.52 for10× 10 mm2 specimens or fvac/fair = 1.28 for 5× 5 mm2 size, respectively.

The membrane stress evaluation was based on the known expression for square membranenatural frequency

f11 =√

22

√τ

µA, (4.32)

where τ is the tension per unit length of edge, A is the area of membrane and µ is the massper unit area. As the value of stress is related to the square of frequency, any deviations of thereal value will cause considerable errors in stress determination. That is why the measurementin vacuum conditions is highly desirable. Regarding this fact a vacuum apparatus was installedwith an optical window for membranes illumination and reflection backward. To excite themembranes vibrations in vacuum an electrostatic loading was implemented.

A number of specimens were analyzed with the tension stresses varying from nearly zero toabout 30 MPa, the corresponding resonant frequencies reached the values up to 20 kHz. Statis-tical assessment and the repeatability obtained allow us to estimate the measurement uncertaintyto about ∆σ = ±0.1 MPa even on small mm sized samples which value seems to be a very goodresult for such a measurements.

Sometimes, either from designing or economical reasons, the use of LDV is not a profitablesolution. That was also the case when the small membranes had to be measured in-situ during theprocess of their irradiation by ions flux. In accordance with this demand an optical autocollima-tion method intended for detection of dynamic membrane bulging immediately inside vacuumchamber has been developed. The autocollimation detector where the object is illuminated bya collimated parallel beam, detects the changes in membrane surface curving. In the layout thelaser diode emits as a point-like source and after its reflection from the mirror-like membranesurface the rays are again collected on the opening of pinhole. Provided that the surface is flat,the properly chosen opening diameter (about Ø = 0.1 mm) ensures all the reflected light is pass-ing through. The periodical bulging of vibrating membrane dissipates the light on the diaphragmscreen thus reducing intensity on the photodiode. After its amplification the useful signal is pro-cessed by FFT and displayed in real time mode for frequency scanning. One of the practicalbenefits of the scheme is the possibility of installing it outside the vacuum chamber at arbitrarylarge working distance from the specimen. The sensitivity of detection is approved to be suffi-cient and moreover the detector can be installed immediately inside the vacuum, then only theelectrical connections are needed. Such detector was used to monitor the degradation changesin thin layer films at the process of ions irradiation by means of continual stress evaluation suc-cessfully. The drawback of the confocal scheme follows from its limitation to measure the firstnatural mode of membrane bulging.

The analysis of vibration spectra of microcomponents and also membranes, as a rule aboundsin uncertainties of main resonant peaks identification. Measuring the vibrations of “unknown”


Beamspliter Point-like

light source

Collimator lens

Membrane

Loud speaker

CCD

Fig. 4.18. Visualization of membranes vibration modes in focal plane of the lens.

peaks of considerable intensity are displayed as harmonic multiples due to non-harmonicallyshaped signals. When the bulging shapes exceed the approximations regularly assumed in ana-lytical solutions, the observation and visualization of membrane shapes can become important.Frequently used manner of mode shapes visualization is the method of scanning LDV or eventu-ally Electronic Speckle Pattern Interferometry (ESPI). Also microinterferometric technique canbe used. Both, the classic interferometry and ESPI can operate either in time-average or in stro-boscopic modes. Recently, we have developed the technique of vibration modes visualizationbased on observation of time-averaged intensity distribution at focal plane of autocollimationscheme. The principal layout is drawn in Fig. 4.18.

Based on the theory of diffraction the intensity distribution can be obtained by solving theFourier transform integral. Provided that the mebrane surface is expressed as a function w(x, y)and the constant distribution of illumination intensity on the membrane area is assumed, complexamplitude at focal plane of image lens will be given simply by Fourier transform of exponentialfunction exp[i2π/λw(x, y)]

a(xf , yf ) =∫∫ ∞

−∞exp

[i2πλw(x, y)

]exp

[−i

2πλf

(xxf + yyf )]

dxdy (4.33)

where the quadratic phase factor in front of the integral has been omitted as irrelevant whensearching for intensity distribution. To find the distribution of intensity field across the focalplane of the lens a squared value of complex amplitude a(xf , yf ) is derived. Furthermore, thetime average mode needs the focal intensity pattern to be integrated in time with regard to theoscillating harmonic movement. As the bulging shapes can be mostly expressed in a simpleanalytical form of harmonic oscillation, the solution of integral either in a closed form or in dis-crete form numerically should be successfully performed. Experimental materialization of theproposed detector has shown that the method provides good conditions for identification of sep-arated vibration modes. In the same way, the vibration at the intervals “between the resonances”is feasible to observe. An example of time-averaged intensity spots for three selected modes isshown in Fig. 4.19 where also in its left part the appropriate bulging shapes are drawn. The pat-terns listed show the bulging of square-shaped membrane 10 × 10 mm2 with maximum centraldeflection in the range of 5 µm. As it can be seen, the extrapolated sensitivity for bulging val-ues detection is comparable or even exceeds the interferometrical measurements. Although timeaveraged, the clear and contrast patterns are obtained with excellent resolution of any membraneshape deviations.


Fig. 4.19. The intensity patterns in the focal plane of the different vibration modes.

Fig. 4.20. The examples of higher order membrane vibration modes obtained by defocusing phase visual-ization.

Moreover, besides the use of time-average focal plane visual display, an optical method hasbeen proposed where image defocusing near the image plane acts as a phase visualization. Obser-vation plane shift of about 10 mm at the projecting lens focal distance f = 1000 mm can detectshape variations of the mirror-like surface. As it was experimentally approved, subnanometermembrane deflections can be detected. Time-average microscopic phase visualization gives riseto characteristic patterns of different vibrating modes (See fig. 4.20). Both schemes are non-interferometric techniques of mode shapes visualization.


5 ConclusionsAddressing a large part of engineering but also scientific tasks cannot be imagined without theuse of the experimental mechanics tools. Although computational methods based on finite ele-ments or boundary elements are widely used today, experimental methods still have their interestin the testing and diagnostics of the deformable body. Significant share of these tools includesmetrological methods using optical principles for monitoring and quantitative evaluation of me-chanical quantities such as mechanical strain, vibration modes, thermal and moisture propagationmeasurement, surface roughness. About the last 50 years, optical methods in experimental me-chanics have passed a long way of development, from the invention of new principles to theirimprovements and application in many areas. Today, optical measurements are still playing im-portant role in the experimental field, although many of these procedures and equipment are nowout-of-date.

Nevertheless, the historical development of the use of different principles, where the keypoint was the discovery of the holographic or speckle interferometry, respectively, may be inmany respects instructive and useful even for a reader not directly interested in the subject-matter.This work more or less follows the mainstream in the last decades and on the basis of author‘sown experience and shows development directions of the fundamental principles based mainlyon the exploitation of coherent light imaging. Apart from this, the physical aspects of imagingby coherent light and creating an interference pattern by light that is coherent per partes within aradius of correlation, are still relevant also for a design of new or improved interferometers.

As a rule, today, wet-processing of photographic material is not used. The design of laserdiode as well as invention of CCD or CMOS matrix in connection with PC enabled immediatedigitalization of the recorded image. Unfortunately, up to date, CCD (CMOS) pixel size is stilltoo rough to record micro-interference pattern. Thus, direct hologram phase registration is prob-lematic and only the speckle field intensity distribution can be analysed. This creates limitationsfor the holographic/speckle records and retards effort to obtain complete information on all theorthogonal components of the displacement vector. Moreover, noticeable limitations result alsofrom interference sensitivity or resolving power of the electronic speckle pattern interferome-try which is often unsufficient to measure small scale deformation. Especially, stress analysisof the deformable body suffers from the necessity to convert information on displacement fieldinto strains by mathematical processing. Mesh of incremental information on the interferencefringes positions is usually too gappy to its direct derivation and high-performance fitting pro-cedures is necessary to use. One of the perspective solution, as we have proposed, is the hybridexperimental-numerical approach, where the primary data obtained from experiment are evalu-ated with respect to wanted physical quantity distribution.

Finally, as it can be concluded against the background of past gradual development, theuse of optical metrology tools has been accepted across the breath of engineering and scientificdisciplines. The use of computers and the photoelectric data collection with the digital imageand electric signal processing, including data fitting procedures, points to the new possibilitiesto enhance performaces and extending the scope of the coherent light imaging for measurementpurposes.

Acknowledgement

This work was supported by the Slovak Research and Development Agency under the contractsNo. APVV-0108-11 and No. APVV-0088-12.

References 215

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Milan Drzık studied the physics at Comenius University in Bratislava.Since 1974 he has been with Slovak Academy of Sciences, now is with In-ternational Laser Centre, Bratislava. He received PhD degree in 1981. Hisresearch interests have been focused on advanced applied optics measuringmethods such as conventional, holographic and speckle interferometry, laserand CCD-based techniques and their applications to solve the problems ofmechanics, investigation of optical, thermophysical and mechanical proper-ties of microstructures.

Juraj Chlpık recieved his PhD degree in Optics and Optoelectronics in 2010at the Faculty of Mathematics, Physics and Informatics, Commenius Uni-versity in Bratislava. He is working as a researcher in International LaserCentre in Bratislava, and since 2008 as an Assistant Professor at the Facultyof Electrical Engineering and Information Technology, Slovak Universityof Technology in Bratislava. His research efforts are focused on apply-ing of optical methods for investigation of material properties of thin lay-ers.

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acta physica slovaca vol. 64 No. 2 & 3, 101 – 216 April ...acta physica slovaca vol. 64 No. 2 & 3, 101 – 216 April & June 2014 OPTICAL METHODS IN EXPERIMENTAL MECHANICS Milan Drzˇ´ık,1∗